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+{"text":"\\section{The SU(N+1) Schwarzschild-like Solution}\n\nIn a previous paper \\cite{sing} we exploited the connection between\ngeneral relativity and Yang-Mills theory to find an exact\nSchwarzschild-like solution for an SU(2) gauge theory coupled to\na massless scalar field. In the present paper we wish to show that\na similiar solution can be found for the general group SU(N+1).\nInstead of using the Euler-Lagrange formalism which leads to coupled\nsecond-order, nonlinear equations, we will use the Bogomolny\napproach \\cite{bogo} to derive our field equations.\nBogomolny  obtained his first-order version of the Yang-Mills\nfield equations by requiring that the gauge and scalar fields\nproduce an extremum of the canonical Hamiltonian. The field equations\nobtained in this way are first-order, but their solutions are\nalso solutions to the second-order Euler-Lagrange equations.\n\nThe model which we consider here is an SU(N+1) gauge field coupled\nto a massless scalar field in the adjoint representation. The\nLagrangian for this theory is\n\\begin{equation}\n\\label{lagran}\n{\\cal L} = -{1 \\over 4} F^{\\mu \\nu a} F_{\\mu \\nu} ^a + {1 \\over 2}\nD^{\\mu} ( \\Phi ^a ) D_{\\mu} ( \\Phi ^a)\n\\end{equation}\nwhere\n\\begin{equation}\nF_{\\mu \\nu} ^a = \\partial _{\\mu} W_{\\nu} ^a - \\partial _{\\nu}\nW_{\\mu} ^a + g f ^{abc} W_{\\mu} ^b W_{\\nu} ^c\n\\end{equation}\nand\n\\begin{equation}\nD_{\\mu} \\Phi ^a = \\partial _{\\mu} \\Phi ^a + g f ^{abc}\nW_{\\mu} ^b \\Phi ^c\n\\end{equation}\nwhere $f^{abc}$ are the structure constants of the gauge group\nand $a, b, c = 1, 2, \\dots , N, N+1$.\nThe canonical Hamiltonian obtained from Eq. (\\ref{lagran}) is\n\\begin{equation}\n\\label{hamo}\n{\\cal H} = \\int d^3 x \\left[{1 \\over 4} F_{ij} ^a F^{aij} - {1 \\over 2}\nF_{0i} ^a F^{a0i} + {1 \\over 2} D_i \\Phi ^a D^i \\Phi ^a\n- {1 \\over 2} D_0 \\Phi ^a D^0 \\Phi ^a \\right]\n\\end{equation}\nWe wish to find gauge and scalar fields which produce\nan extremum of ${\\cal H}$.\nFirst we rescale the scalar field ({\\it i.e.} $\\Phi ^a \\rightarrow\nA \\Phi ^a$). This is done so that later on it will be simple to examine\nthe pure gauge case by setting $A=0$.\nNext we specify that all the fields are time independent, and that\nthe time component of the gauge fields are proportional to the\nscalar fields ({\\it i.e} $W_0 ^a = C \\Phi ^a$, where $\\Phi ^a$ is\nthe rescaled scalar field). The time component of the gauge fields\nact like an additional Higgs field except its kinetic term appears\nwith the opposite sign in the Lagrangian \\cite{zee}. Using these two\nrequirements and the antisymmetry of $f^{abc}$ we find that\n$D_0 \\Phi ^a = 0$ and $F_{0i} ^a = C (D_i \\Phi ^a)$, so that the\nHamiltonian becomes\n\\begin{eqnarray}\n\\label{ham}\n{\\cal H} = \\int d^3 x \\Big[&&{1 \\over 4} \\left(F_{ij} ^a - \\epsilon_{ijk}\n\\sqrt{A^2 - C^2} D^k \\Phi ^a \\right) \\left(F^{aij} - \\epsilon_{ijl}\n\\sqrt{A^2 - C^2} D^l \\Phi ^a \\right) \\nonumber \\\\\n+ && {1 \\over 2} \\epsilon_{ijk} \\sqrt{A^2 - C^2}\nF^{aij} D^k \\Phi ^a \\Big]\n\\end{eqnarray}\nUsing the fact that\n\\begin{equation}\n{1 \\over 2} \\epsilon_{ijk} F^{aij} D^k \\Phi ^a = \\partial ^i\n\\left({1 \\over 2} \\epsilon_{ijk} F^{ajk} \\Phi ^a \\right)\n\\end{equation}\nand the requirement that the solutions we are looking for are\nonly functions of $r$ we find\n\\begin{eqnarray}\n\\label{ham1}\n{\\cal H} &=& \\sqrt{A^2 - C^2} \\int _S (\\Phi^a B^a _i)\ndS ^i \\nonumber \\\\\n&+& \\int d^3 x \\left[{1 \\over 4} \\left(F_{ij} ^a -\n\\epsilon_{ijk} \\sqrt{A^2 - C^2} D^k \\Phi ^a \\right)\n\\left(F^{aij} - \\epsilon_{ijl} \\sqrt{A^2 - C^2}\nD^l \\Phi ^a \\right) \\right]\n\\end{eqnarray}\nFor the total divergence term we\nhave used the definition of the non-Abelian\nmagnetic field in terms of the field strength tensor ({\\it i.e}\n$B^a _i = {1 \\over 2} \\epsilon_{ijk} F^{ajk}$), and used\nGauss's Law to turn the volume integral into a surface\nintegral. The lower limit of this Hamiltonian can be found by\nrequiring\n\\begin{eqnarray}\n\\label{fbogo}\nF_{ij} ^a &=& \\epsilon _{ijk} \\sqrt{ A^2 - C^2}\nD^k \\Phi ^a \\nonumber \\\\\nor \\nonumber \\\\\nB_i ^a &=& \\sqrt{A^2 - C^2} D_i \\Phi ^a\n\\end{eqnarray}\nTo get the second expression we have again used the definition of\nthe non-Abelian magnetic field. These are the Bogomolny equations\n\\cite{bogo}. Wilkinson and Goldhaber \\cite{gold} have given a\ngeneralized ansatz for the gauge and scalar fields\n\\begin{eqnarray}\n\\label{ansatz}\n{\\bf W}_i &=& {\\epsilon_{ijb} r^j ({\\bf T}^b - {\\bf M}^b (r)) \\over g r^2}\n\\nonumber \\\\\n\\Phi ^a &=& {{\\bf \\Phi} (r) \\over g}\n\\end{eqnarray}\n${\\bf W} _i$ are three $(N+1) \\times (N+1)$ matrices of the gauge\nfields. ${\\bf M}_b (r)$ and ${\\bf \\Phi} (r)$ are four $(N+1) \\times\n(N+1)$ matrices whose elements are functions of $r$, and in terms of\nwhich the Bogomolny equations will be written. ${\\bf T}_b$ are three\n$(N+1) \\times (N+1)$ matrices which generate the maximal embedding\nof SU(2) in SU(N+1). Because of the spherical symmetry requirement\none can look at Eq. (\\ref{fbogo}) along any axis \\cite{bais}\n\\cite{wilk}. Taking the positive $\\hat {z}$ axis the Bogomolny\nequations become $\\sqrt{ A^2 - C^2} (D_3 \\Phi ^a) = B _3 ^a$ and\n$\\sqrt{A^2 - C^2} (D_{\\pm} \\Phi ^a) = B_{\\pm} ^a$, or in terms of the\nansatz of Eq. (\\ref{ansatz})\n\\begin{eqnarray}\n\\label{fbogo1}\nr^2 \\sqrt{A^2 - C^2} {d {\\bf \\Phi} \\over dr} &=&\n[{\\bf M}_+  , {\\bf M}_- ] - {\\bf T}_3  \\nonumber \\\\\n{d {\\bf M}_{\\pm} \\over dr} &=& \\mp \\sqrt{A^2 - C^2}\n[{\\bf M}_{\\pm} , {\\bf \\Phi} ]\n\\end{eqnarray}\nTaking the third ``component'' of the maximal SU(2) embbedding into\nSU(N+1) as\n\\begin{equation}\n{\\bf T}_3 = diag \\left[{1 \\over 2}N, {1\\over 2}N -1, \\dots ,\n-{1 \\over 2} N +1, -{1 \\over 2} N \\right]\n\\end{equation}\nit has been shown \\cite{gold} that the matrix functions, ${\\bf M}_+ (r)$\nand ${\\bf \\Phi} (r)$, can be taken as\n\\begin{eqnarray}\n\\label{mat1}\n{\\bf \\Phi} = {1 \\over 2}\n\\left(\n\\begin{array}{ccccc}\n\\phi_1 &\\space &\\space &\\space &\\space \\\\\n\\space &\\phi_2 - \\phi_1 &\\space &\\space &\\space \\\\\n\\space &\\space &\\ddots &\\space &\\space \\\\\n\\space &\\space &\\space &\\phi_N -\\phi_{N-1} &\\space \\\\\n\\space &\\space &\\space &\\space &- \\phi _N \\\\\n\\end{array}\n\\right)\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{mat2}\n{\\bf M}_+ = {1 \\over \\sqrt{ 2}}\n\\left(\n\\begin{array}{ccccc}\n0 &a_1 &\\space &\\space &\\space \\\\\n\\space &0 &a_2 &\\space &\\space \\\\\n\\space &\\space &\\ddots &\\space &\\space \\\\\n\\space &\\space &\\space &0 &a_N \\\\\n\\space &\\space &\\space &\\space &0 \\\\\n\\end{array}\n\\right)\n\\end{eqnarray}\nwhere $\\phi _m$ and $a_m$ are real functions of $r$ and ${\\bf M}_- =\n({\\bf M} _+) ^T$. Substituting Eqs. (\\ref{mat1}), (\\ref{mat2}) into\nthe first order field equations of Eq. (\\ref{fbogo1}) the field\nequations become \\cite{bais} \\cite{wilk}\n\\begin{eqnarray}\n\\label{deqn}\nr^2  {d \\phi _m \\over dr} &=& {1 \\over \\sqrt{A^2 - C^2}}\n\\left[ (a_m)^2 - m{\\bar m} \\right] \\nonumber \\\\\n{d a_m \\over dr} &=& \\sqrt{A^2 - C^2} \\left(-{1 \\over 2} \\phi_{m-1}\n+\\phi_m - {1 \\over 2} \\phi _{m+1} \\right) a_m\n\\end{eqnarray}\nwhere $1 \\le m \\le N$, ${\\bar m} = N +1 - m$ and $\\phi_0 = \\phi_{N+1} =0$.\nExact solutions have been found to Eq. (\\ref{deqn}) \\cite{bais}\n\\cite{wilk} which are generalizations of the well known\nPrasad-Sommerfield solution \\cite{prasad} for SU(2). The\nPrasad-Sommerfield solution and their generalizations satisfy\nthe boundary condition that the gauge and scalar fields are\nfinite at the origin. If one does not require that the fields\nbe finite at the origin then the coupled equations for $\\phi _m (r)$\nand $a_m (r)$ can be solved by\n\\begin{eqnarray}\n\\label{soln}\n\\phi _m (r) &=& {1 \\over \\sqrt{A^2 - C^2}}\n{K m {\\bar m} \\over r (K - r)} \\nonumber \\\\\na_m (r) &=& {r \\sqrt{m {\\bar m}} \\over K - r}\n\\end{eqnarray}\n$K$ is an arbitrary constant with the dimensions\nof distance. This is the generalization of a similiar\nsolution which we found for SU(2) using the second-order\nEuler-Lagrange formalism \\cite{sing}. The reason for wanting to\ngeneralize our solution to SU(N+1) is to give a possible explanation for\nthe confinement mechanism in QCD whose gauge group is SU(3). Inserting\nthe functions $\\phi _m (r)$ and $a_m (r)$ of Eq. (\\ref{soln}) into\nthe fields of Eq. (\\ref{ansatz}) we find that these fields become\ninfinite at a finite radius of\n\\begin{equation}\nr_0 = K\n\\end{equation}\nThe non-Abelian ``electric'' ($E_i ^a = F_{0i} ^a$) and ``magnetic''\nfields ($B_i ^a = {1 \\over 2} \\epsilon_{ijk} F^{jka}$) calculated for\nthis solution also become infinite at $r_0$. Thus any particle which\ncarries an SU(N+1) charge would either never be able to penetrate beyond\n$r_0$ (when the SU(N+1) charges are replusive) or once it passed into\nthe region $r < r_0$ it would  never be able to escape  back to the\nregion $r > r_0$. One has in a sense a color charge black hole.\n\nBoth the present solution and our SU(2) solution were found by\nusing the connection between Yang-Mills theory and general relativity\n\\cite{utiyama}, and trying to find the Yang-Mills equivalent of the\nSchwarzschild solution. The objects in general relativity which correspond\nto the gauge fields are the Christoffel coefficients, $\\Gamma^{\\alpha}\n_{\\beta \\gamma}$. Examining a few of the Christoffel symbols of the\nSchwarzschild solution we find\n\\begin{eqnarray}\n\\Gamma ^t _{r t} &=& {2GM  \\over 2r ( r - 2GM )} \\nonumber \\\\\n\\Gamma ^r _{r r} &=& -{2GM  \\over 2r ( r - 2GM )}\n\\end{eqnarray}\nwhere $2GM$ the equivalent of the constant $K$ from the Yang-Mills\nsolution. The similarity between these Christoffel coefficients and the\ngauge and scalar fields that result from the solutions, $\\phi _m (r)$\nand $a_m (r)$ of Eq. (\\ref{soln}), is striking. The most important\nsimilarity from the point of explaining confinement is the existence\nin both solutions of an event horizon, from which particles which carry\nthe appropriate charge can not escape once they pass into the region\n$r < r_0$. For general relativity the appropriate ``charge'' is\nmass-energy so that nothing can climb back out of the Schwarzschild\nhorizon, while in the Yang-Mills case only particles carrying an SU(N+1)\ncharge will become confined.\n\nOne slightly disturbing feature of these Schwarzschild-like SU(N+1)\nsolutions is that they have an infinite energy due to the\nsingularity at $r = 0$.\nWhen quantities such as the energy of this field configuration\nare calculated the integral must be cutoff at some\narbitrary radius, $r_c$. The singularity at\n$r_0$ does not give an infinite energy unless one sets\n$r_c = K$. This singular behaviour at the origin is shared\nby several other classical field theory solutions. Both the\nSchwarzschild solution of general relativity, and the Coulomb solution\nin electromagnetism have similiar singularities.  The Wu-Yang\nsolution \\cite{wu} for static SU(2) gauge fields with no time component\nalso blows up at the origin, leading to an infinite energy if one\nintegrates the energy density down to $r=0$. Just as these classical\nsolutions are not expected to hold down to $r=0$ so the present solution\nwill certainly be modified by quantum corrections as $r$ approaches\nzero. Phenomenologically we know that the present solutions can not\nbe correct for very small $r$, since they do not exhibit the asymptotic\nfreedom behaviour that is a desirable consequences of the quantum\ncorrections to QCD. It would be interesting to see if the behaviour of\nthe fields at the origin could be modified by introducing a mass\nterm (${m^2 \\over 2} \\Phi ^a \\Phi ^a$) and a self interaction term\n(${\\lambda \\over 4} (\\Phi ^a \\Phi ^a) ^2$) to the scalar field part of\nthe Lagrangian, while still retaining the color event horizon feature of\nthe present solution. This smoothing of the fields at the origin does\nhappen when one compares the Prasad-Sommerfield exact solution (where\nthere are no mass or self interaction terms for the scalar field) with\nthe the numerical results of 't Hooft \\cite{thooft} or Julia and Zee\n\\cite{zee} (where mass and self interaction terms are included).\nThe numerical results lead to monopoles and dyons with a\nfinite core, while the exact Prasad-Sommerfield solution leads to a\npoint monopole (despite this their exact solution still has finite\nenergy when the energy density is integrated down to zero, unlike our\npresent solution). As in the case of the Prasad-Sommerfield solution,\nintroducing mass and self interaction terms for the scalar fields\nwould require solving the equations numerically, since we have\nnot been able to find an analytical solution under these conditions.\n\nTo calculate the energy of the field configuration of our solution it is\nnecessary to integrate the $T_{00}$ component of the energy-momentum\ntensor  over all space, excluding the origin. The  $T_{00}$\ncomponent of the energy-momentum tensor\nis similiar to the Hamiltonian density of Eq. (\\ref{ham}) except that\nall the terms have positive signs\n\\begin{equation}\nT_{00} = {1 \\over 4} F_{ij} ^a F^{aij} + {1 \\over 2} F_{0i} ^a\nF^{a0i} + {A^2 \\over 2} D_i \\Phi ^a D^i \\Phi ^a  + {A^2 \\over 2}\nD_0 \\Phi ^a D^0 \\Phi ^a\n\\end{equation}\nThe energy in the fields of our solution is the integral over all\nspace of $T_{00}$. Using the field equations of Eq. (\\ref{fbogo}),\nand the fact that $F_{0i} = C(D_i \\Phi ^a)$ the energy of the fields\nis\n\\begin{eqnarray}\nE &=& \\int T_{00} d^3 x \\nonumber \\\\\n&=& A^2 \\int d^3 x D_i \\Phi ^a D^i \\Phi ^a\n\\end{eqnarray}\nSince the fields only have a radial dependence the angular part\nof the integration can be easily done. Futher using the radial\nsymmetry to evaluate the integrand along the positive $\\hat{ z}$\naxis, and using the matrix expression for $\\Phi ^a$ as well as the\nfield equations, Eq. (\\ref{fbogo1}), we find that the energy\nbecomes \\cite{gold}\n\\begin{equation}\n\\label{energy1}\nE = {8 \\pi A^2 \\over g^2} \\int _{r_c} ^{\\infty} r^2 dr \\left(\nTr \\left( \\left[{d {\\bf \\Phi } \\over dr} \\right] ^2 \\right)\n+ {2 \\over \\ A^2 - C^2} Tr \\left( r^{-2} {d {\\bf M_+} \\over dr}\n{d {\\bf M_-} \\over dr} \\right) \\right)\n\\end{equation}\nUsing the solutions for the elements of the matrices,\n$\\bf{ \\Phi}$ and $\\bf{ M}_{\\pm}$ of Eq. (\\ref{soln}) we find\n\\begin{eqnarray}\n\\label{phim}\nTr \\left( \\left[ {d {\\bf \\Phi } \\over dr} \\right]^2 \\right)\n&=& {K^2 (2r - K)^2 \\over 4 r^4 (A^2 - C^2)\n(K - r)^4} \\sum _{n=0} ^N (N - 2n)^2 \\nonumber \\\\\nTr \\left( r^{-2} {d {\\bf M_+} \\over dr} {d {\\bf M_-} \\over dr} \\right)\n&=& {K^2 \\over r^2 (K-r)^4} \\sum _{n=0} ^N n(N+1-n)\n\\end{eqnarray}\nUsing this in Eq. (\\ref{energy1}) and carrying through the integration\nthe energy in the field configuration of this Schwarzschild-like\nsolution is\n\\begin{equation}\n\\label{energy2}\nE = {2 \\pi A^2 K^2 N(N+1)(N+2) \\over 3 g^2 (A^2 - C^2)}\n\\left[ {K  - 2 r_c \\over r_c (K -r_c)^3} \\right]\n\\end{equation}\nwhere the sums from Eq. (\\ref{phim}) have been done explicitly.\nThis result can be checked against the SU(2) result \\cite{sing}\nby taking  $N=1$ in Eq. (\\ref{energy2}), and the expressions for\nthe energy do indeed agree (In Ref. \\cite{sing} we required that\n$A^2 - C^2 =1$ whereas here this factor is divided out). As it\nstands there are two arbitrary constants that enter the solution\n({\\it i.e.} $K$ and $r_c$) which would have to be specified\nbefore any connection between this Schwarzschild-like solution\nand the real world could be carried out. As has already been\nmentioned $K$ is the Yang-Mills equivalent of $2GM$ in general\nrelativity. Thus it could be conjectured that $K$ is related\nto the strength of the gauge interaction\n($G$ in general relativity), and the\nmagnitude of the central charge which produces the gauge field\nconfiguration ($M$ in general relativity).\n\nOne interesting feature which this general SU(N+1) Schwarzschild-like\nsolution shares with our previous SU(2) solution is that scalar\nfields are apparently required in order to get a physically\nnon-trivial solution. If there where no scalar fields in the original\nLagrangian ({\\it i.e.} $A = 0$), then the field energy of Eq.\n(\\ref{energy2}) would be zero, and the $W_0 ^a$ component of the\ngauge fields would be pure imaginary. Although the pure gauge\ncase with no scalar fields is a solution mathematically, its\nphysical significance is dubious. Requiring that the\nsolutions are pure real, or that the energy in the fields\nbe non-zero would exclude the pure gauge case solution. Under either\nof these requirements on the solution, it can be seen that scalar\nfields must be present.\n\n\\section{Conclusions}\n\nIn this paper we have generalized our previous exact,\nSchwarzshild-like solution for SU(2) Yang-Mills theory\nto SU(N+1) by using an embedding of SU(2) into SU(N+1) \\cite{gold}.\nThis exact SU(N+1) solution was found by using the connection\nbetween general relativity and Yang-Mills theories. It was found that\nthe Schwarzschild solution of general relativity carries over\nwith only a little modification into an equivalent solution for an\nSU(N+1) gauge theory coupled to massless scalar fields. Just as the\nSchwarzschild solution possesses an event horizon which permanently\nconfines any particle which carries the ``charge'' of the  gravitational\ninteraction ({\\it i.e} mass-energy), so the present solution also\nhas a ``color'' event horizon which permanently confines any particle\nwhich carries the SU(N+1) gauge charge. This may be the confinement\nmechanism which has long been sought for the SU(3) gauge theory of\nthe strong interaction, QCD. Before this claim can be made there\nare several important questions which must be resolved. First, under\nsome reasonable physical assumptions about the nature of our solution\nit is found that scalar fields are required for a solution to\nexist. Normally scalar fields are not thought to play a significant role\nin confinement, so the physical importance of these scalar fields\nwould need to be addressed. Second, there are several arbitrary constants\nwhich crop up in the solution ($K$ and $r_c$). In order to make a\nconnection with the real world these constants would have to be given.\nTheoretically $K$ should be related to the strength of the gauge\ninteraction as well as the magnitude of the gauge charge which produces\nthe Schwarzschild-like gauge fields. Experimentally $K$ should be\nrelated to the radius of the various QCD bound states ({\\it e.g.}\nprotons, pions, etc.). The other constant, $r_c$, was\nintroduced chiefly to avoid the\nsingularity at $r=0$, but also because our solution does not\npossess the property of asymptotic freedom as $r \\rightarrow 0$.\nThis should not be too surprising since our solution is\nfor classical Yang-Mills fields, but as $r \\rightarrow 0$ quantum\neffects should become increasingly important.\nThus $r_c$ can be thought of roughly, as marking the boundary between\nthe classical, confining solution of this paper, and the quantum\ndominated asymptotic freedom regime. All this strongly suggests\na bag-like structure for QCD bound states : As a particle approaches\n$r = K$ from $r < K$, it feels a progressively  stronger color force\nwhich confines it to remain inside the bound state. As the particle\napproaches $r \\rightarrow 0$ it enters the asymptotic freedom\nregime, where it moves as if it were free.\n\nAn interesting extension of this work would be to see if other\nexact solutions from general relativity have Yang-Mills counterparts.\nIn particular if a Yang-Mills equivalent of the Kerr solution\ncould be found it might give some insight into the nature of\nthe spin of fermions.\n\n\\section{Acknowledgements}  I would like to acknowledge the help\nand suggestions of David Singleton and Hannelore Roscher.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nStarting with the original Great Dark Spot (GDS-89) observed by {\\it Voyager 2}, roughly a half-dozen large geophysical vortices have been observed on the Ice Giants. In 2015-2017, a feature was observed on Neptune in the southern hemisphere (SDS-2015) \\citep{Wong2016,Wong2018}. A recent observation as part of the Hubble Outer Planet Atmosphere Legacy (OPAL) program revealed a new dark spot with bright companion clouds in Neptune's northern hemisphere, NDS-2018 \\citep{Simon2019}. The structure is the most recent of large scale geophysical features\nto be observed on the ice giants. Although these Dark Spots have similar features to large vortices on Jupiter, such as the Great Red Spot, they exhibit dynamical motions such as shape oscillations and latitudinal drift. Neptune anticyclones evolve on a time scale of months \\citep{hsu2019} as opposed to large vortices on Jupiter, which evolve over many decades \\citep{Ingersoll2004}.  During the {\\it Voyager 2} encounter with Neptune beginning in January of 1989, GDS-89 was drifting towards the equator at an approximate rate of 1.3\\degree \/month \\citep{Sromovsky2002}. Post-{\\it Voyager 2} observations using the Hubble Space Telescope (HST) revealed that the vortex had dissipated as it approached the equator \\citep{HammelLockwood1997}. Many of the observed characteristics have been matched previously by numerical models\n(e.g., shape oscillation by \\citet{Polvani1990}, drift rate by \\citet{LeBeauDowling_Neptune} and companion cloud formation by \\citet{Stratman2001}). However, reproducing all of these features simultaneously has remained elusive.\n\nThese vortices exhibit surprising variability in terms of evolution, shape, drift, cloud distribution, and shape oscillations, so an explicit calculation of the environmental parameters is required for each vortex observed. However, this is beneficial because more diagnostic information can be obtained from each case than if the spot occurrences were repetitive in terms of their characteristics. For example, in the case of GDS-89, equatorial drift was observed \\citep{Smith1989,LeBeauDowling_Neptune} as opposed to the poleward drift of dark spot SDS-2015 \\citep{Wong2016,Wong2018}. In addition, GDS-89 had an accompanying cloud feature external to the dark spot, presumably condensed methane, whereas some vortices have centered bright companions such as D2 \\citep{Smith1989} and SDS-2015 \\citep{Wong2018}. \n\n\\citet{Stratman2001} demonstrated that orographic upwelling could be the cause of companion clouds. An analysis of these overlying orographic cloud features shows promising insight into the dynamics of clouds on Neptune, as the features are directly impacted by the underlying atmospheric structure and the deep methane abundance. To that end, it is necessary to apply a cloud microphysical model to make a more complete representation of dark spots. We use an updated microphysics calculation implemented in the Explicit Planetary Isentropic Coordinate General Circulation Model (EPIC GCM) \\citep{Dowling_EPICmodel1998,Dowling2006,Palotai2008} to account for methane cloud microphysics (see Section~\\ref{clouds}). We investigate the dynamics of vapor and subsequent persistent cloud formation on Neptune by modelling large scale vortices. \n \nClouds have been modelled explicitly in recent works to study the effect of convection \\citep[e.g.,][]{Sugiyama2011,Li2019} and the GRS \\citep{Palotai2014} on Jupiter, but not for Neptune. \n\n\nWe seek to answer the following questions: \n\\begin{enumerate}\n    \\item How does the addition of a methane cloud microphysical model affect the evolution of the vortex? \n    \\item How do various methane deep abundance (mole fraction) values affect the vortex? \n\\end{enumerate}\n\nTo investigate these questions, we use an increased horizontal grid resolution and increased number of vertical layers in the column compared to previous studies. \nAdditionally, we use methane vapor to track the vortex drift rate and shape oscillations whereas in previous studies, potential vorticity (PV) was used as the primary parameter to determine the features of the vortex.\n\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{Figures\/neptune_voyager.jpg}\n    \\caption{An image of the GDS-89 (dark feature near the center) taken by {\\it Voyager 2} in 1989. The high altitude white companion clouds around the GDS are visible. To the South, the Scooter, which is another dark spot observed by {\\it Voyager 2} is visible with its distinct centered cloud. Courtesy NASA\/JPL-Caltech.}\n    \\label{fig:GDS-89}\n\\end{figure}\n\n\\section{Methods}\n\\subsection{EPIC model}\nTo investigate persistent cloud formation on Neptune, we use the EPIC-GCM with an active hydrological cycle for methane.\nEPIC uses a hybrid vertical coordinate, $\\zeta$, described by \\citet{Dowling2006}. The top of the model uses potential temperature, $\\theta$, as a vertical coordinate while the bottom uses a scaled pressure variable, $\\sigma$. The transition between the two functions occurs at 10 hPa (see Figure \\ref{fig:TP}). Indeed, this transition allows for the use of EPIC on planets without a solid, terrestrial surface (i.e. gas giants) and increased vertical resolution at deeper layers where $\\theta$ is nearly constant. \n\n\\subsection{Cloud microphysics} \\label{clouds}\nWe use the active cloud microphysical model from \\cite{Palotai2008} which incorporates the condensation of methane in the EPIC model, with the revisions to precipitation given in \\cite{Palotai2016DPS}. \n\nThis scheme deals with bulk mass transfer between five explicit phases: vapor, cloud ice, liquid cloud droplet, snow and rain. The last two correspond to precipitation in the model and are subject to sedimentation at the terminal velocity.\n\nCloud particle sizes are diagnosed using the Gunn-Marshall distribution \\citep{GunnMarshall} for snow and the Marshall-Palmer distribution \\citep{MarshallPalmer} for rain. Both are log-normal with two free parameters describing the mean and standard-deviation of the particle-size distribution. We apply these distributions due to the low number of parameters (2) and the computational efficiency they provide compared to other, more recently retrieved parameterizations. Furthermore, most Earth-based literature use empirically derived distributions which are accurate for localised conditions, making them a poor choice for porting to gas-giant atmospheres \\citep{Palotai2008}.\n\n\\subsection{Fall Speed of Particles}\n\n\nTo speed up computation, we parameterize the terminal velocity snow particles of different diameters ($D$) using a power law:\n\\begin{equation}\\label{eq:fall_speed}\n    V_t = x D^y \\left( \\dfrac{p_0}{p}\\right)^{\\gamma}\n\\end{equation}\nwhere $x$, $y$ and $\\gamma$ are determined by fitting the terminal velocity calculated using  theoretical hydrodynamical principles  \\citep{PruppacherKlettBookFallSpeed}. $p_0=1$ bar is the reference pressure. Similarly to \\citet{Palotai2008}, we assume that snow is graupellike with a hexagonal shape and follow their formulation for calculating sedimentation velocity as a function of particle size. \n\nA plot of the terminal velocity of CH$_4$ and H$_2$S snow at a pressure of 1 bar is shown in Figure~\\ref{fig:termvel}, and the fit parameters for CH$_4$ and H$_2$S on Neptune are given in Table \\ref{tab:fall_speed}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/neptune_snow.png}\n    \\caption{Snow terminal velocity fits for CH$_4$ (solid) and H$_2$S (dashed) at 1 bar on Neptune.}\n    \\label{fig:termvel}\n\\end{figure}\n\nUsing the particle size distributions, the net sedimentation rate for a grid cell is calculated by weighting the terminal velocities by the mass of particles in each radius bin. Currently, only snow particles undergo sedimentation since the terminal velocity for cloud ice on Neptune is on the order of a few $\\mu$m\/s and thus fall only a few meters over the duration of our simulations.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\centering\n  & CH$_4$ snow & H$_2$S snow  \\\\ \\hline \\hline\n$x$         & 14.5   & 22.2    \\\\ \\hline\n$y$         & 0.458  & 0.504   \\\\ \\hline\n$\\gamma$    & 0.320  & 0.320   \\\\ \\hline\n\\end{tabular}\n\\caption{These constants are determined from Eq. \\ref{eq:fall_speed} when $D$ is in meters and $V_t$ is in m\/s.}\n\\label{tab:fall_speed}\n\\end{table}\n\n\n\n\\subsection{Model setup} \\label{Environmental Initialization}\n\nWe run 3-dimensional simulations that span $-90\\degree$ to $0\\degree$ latitude and $-120\\degree$ to $120\\degree$ longitude with 256 points each resulting in horizontal boxes that are $0.5\\degree\\times1\\degree$ (lat$\\times$lon). These limits are sufficient to minimize the effects of the lateral boundaries on the simulated vortex and its associated clouds and prevent the vortex from interacting with itself \\citep{LeBeauDowling_Neptune}. The model covers 35 unequally spaced vertical layers between $1$ hPa and $14$ bar (see Figure \\ref{fig:TP}), which provides significantly higher resolution than previous studies of the GDS.\n\n\nThe pressure-temperature profile used in this study is shown in Figure \\ref{fig:TP}, which is an idealized curve taken from \\citet{LeBeauDowling_Neptune}. The black curve is from previous refinements of {\\it Voyager 2} radio-occultation T(P) data from \\citet{Conrath1991} and \\citet{Stratman2001} took helium at a 19\\% mole fraction. This profile is used as the initial input into the model at the equator. The temperature profile throughout the rest of the model is calculated using the thermal wind equation. The layers, shown on Figure \\ref{fig:TP} as horizontal lines on the right, were chosen in order to increase the resolution around the initial location of the spot (approximately 1 bar) in the vertical column and the methane cloud deck.  \nIn addition, since the model reaches up to 1 hPa, we can set the $\\sigma-\\theta$\ntransition at 10 hPa, or between $k = 2$ and $k = 3$ indicated by the intersection of the horizontal red line and the potential temperature profile in blue. We initialize equilibrium simulation using the idealized zonal wind profile $Q_y = \\frac{1}{3}$ from \\citet{LeBeauDowling_Neptune}\n, which closely matches the drift rate of the GDS-89 as opposed to other idealized curves (see Fig. \\ref{fig:zonal_wind} and Section \\ref{Drift_rate}). We run 130 day simulations to investigate stability, drift rate, and companion clouds throughout the model, with the ultimate goal of achieving similarities to the observations of GDS-89 and other vortices. Given that the observed periodicity in the oscillation of the GDS-89 was about 8 days \\citep{Sromovsky1993}, the model should be able to capture several periods of wobble in the vortex.\n\nInitially, the geopotential heights are determined in the model by integrating the hydrostatic equation. Due to numerical errors, the model is initially slightly unstable, so we run the model for 5 days to let the winds stabilize. During this time, cloud microphysics is turned off so as to not have spurious cloud growth. The spot is added to the model at the end of this equilibrium phase. \n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{Figures\/tp_profile5.png}\n    \\caption{(a) shows the temperature and pressure profile of Neptune used in the EPIC model. (b) shows the potential temperature profile ($\\theta$) of Neptune (shown in blue). The horizontal black lines indicate the position of vertical layers used in the model. The GDS-89 extends from approximately 200 hPa to  3.5 bars (shown in the shaded region). The sigma-theta transition is also shown in red at 10 hPa.}\n    \\label{fig:TP}\n\\end{figure}\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{Figures\/zonal_wind.png}\n    \\caption{Mean zonal wind profile as a function of latitude using the pseudo PV gradient, $Q_y=\\frac{1}{3}$. The vortex is induced at roughly $32\\degree$ S latitude. }\n    \\label{fig:zonal_wind}\n\\end{figure}\n\n\n\\subsection{Methane abundance}\nThe primary goal of this work is to analyse the effect of methane on the dynamics and stability of the GDS-89. To that end, we vary the deep abundance (mole fraction) of carbon and initial ambient humidity in our test cases. \n\nIn the nominal case, we take the standard value of $40\\times$ the solar [C\/H] fraction for Neptune \\citep{Baines1995}, or an approximate CH$_4$ mole fraction of $\\textit{f}_{\\text{CH}_4} = 0.022^{+0.005}_{-0.006}$. The vertical methane abundance profile is initialized by a ``cold-trap\" model, where the mole fraction at the deepest layer is defined by a constant value (the deep abundance). Successive layers are limited by the minimum of the saturation mole fraction and the mixing ratio of the layer below. In our model, we take the initial relative humidity for all ``wet\" cases to be 95\\% to prevent spurious cloud growth in the first timestep. In a 3-dimensional model, advection of methane vapor and cooling produce localized supersaturation, which leads to cloud formation.\n\nWe run 5 cases which are shown in Table~\\ref{Cases}. Three include active cloud microphysics at different methane deep abundances. In the passive case, methane vapor is added to the model but does not condense. We also include a dry (H\/He atmosphere) case. These different parameters test both the effect of the additional mass from methane vapor and the latent heat release from cloud formation on the dynamics of the GDS. \n\n\\begin{table}\n\\centering\n\\begin{tabular}{ccc}\n\\hline\n\nCase    & Cloud microphysics & [C\/H] relative to solar \\\\ \\hline\\hline\n1       & On                 & $20$                                    \\\\ \\hline\n2       & On                 & $40$                                    \\\\ \\hline\n3       & On                 & $80$                                    \\\\ \\hline\n4       & Off                & $40$                                    \\\\ \\hline\n5       & Off                & -                                       \\\\ \\hline\n\\end{tabular}\n\\caption{A summary of cases tested in this study. For case 4, methane was added but cloud microphysical processes were turned off (i.e. methane vapor was subject only to advection). Case 5 is a dry model which has no added methane.}\n\\label{Cases}\n\\end{table}\n\n\\subsection{Addition of the Vortex} \\label{sec:addvort}\n\nWe utilize the vortex initialization as detailed in \\citet{LeBeauDowling_Neptune}, who use a Kida vortex model.\nThe vortex begins at a latitude of $32\\degree$ S at a pressure level of 1000 hPa ($k = 24$). The vortex extends roughly a scale height vertically in both directions (see Fig. \\ref{fig:TP}). The initial vortex adjustment period is about $6-8$ model days during which the shape of the spot changes drastically. As such, we begin our analysis after this adjustment period; $10$ days into the model. \n\nThe induction of the vortex creates a region of anomalous potential vorticity (PV) compared to the ambient zonal wind shear, defined as the difference in PV between the test case and the corresponding equilibrium simulation. PV has been used in previous modelling works \\citep[e.g.][]{LeBeauDowling_Neptune, Stratman2001} to study the GDS and is a useful tracer in dry models to track the vortex. We can define the extent of the GDS by a contour of constant anomalous PV at a pressure level of $1$ bar or $500$ hPa (the centre of initialized vortex). The shape and location of the vortex are then quantified by fitting an ellipse to these contours using the method of \\citet{HalirFlusser1998}. \n\n\nHowever, the PV of the simulated vortex is difficult to compare with {\\it Voyager 2} observations due to the lack of precise wind speed measurements. The change in the atmospheric structure by the addition of the vortex leads to a distinct variation in methane abundance inside the GDS, which can be used to track the dynamics of the spot in a similar way to PV and can also be compared with the observed spot opacity. By integrating the methane vapor density with depth, we obtain the column density (CD), which is analogous to optical depth in the upper troposphere. We then take the ratio of the CD to the equilibrium (see Section \\ref{Environmental Initialization}) and proceed similarly, by fitting an ellipse to a contour of constant CD. Hereafter, references to CD implies this ratio, unless otherwise specified.\n\n\n\n\n\\section{Results}\n\n\\subsection{1D Simulations}\nWe first run a 1-dimensional case with methane cloud formation enabled to test the microphysical processes and input parameters. We initialize the atmosphere with 150 layers between 1 hPa and 14 bar. We use a nominal value of $40\\times$ the solar [C\/H] fraction, and the model is initially saturated at 100\\% relative humidity everywhere. \n\nFrom equilibrium cloud condensation models (ECCM) \\citep[e.g.,][]{AtreyaWong2005}, methane ice clouds are predicted to form at around 15 bar. Figure~\\ref{fig:1dsim} shows the cloud as a function of time. The base of the clouds in the model are at about 1050 hPa. Snow is shown with the black contours and is precipitated out in a few hours. Cloud ice is most dense near the base and has a typical density of $\\sim 10^{-5}$ kg\/m$^3$. \n\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{Figures\/cloud_dens.png}\n    \\caption{Results from 1D simulations. Methane ice cloud is plotted in light blue and snow is plotted in black dashed contours. Colorbar corresponds to the base-10 log of the cloud density in kg\/m$^3$. }\n    \\label{fig:1dsim}\n\\end{figure}\n\n\\subsection{Potential vorticity} \\label{vorticity_dynamics}\nTo fit the vortex, we use the anomalous potential vorticity as a tracer. We also tracked the evolution of the anomalous PV within the vortex, as shown in Figure~\\ref{fig:pv_evolution}. As the vortex drifted equatorward, the background wind shear and the background potential vorticity increased. Consequently, although the total PV of the vortex itself was conserved, the anomalous potential vorticity of the vortex decreased with time, making it difficult to consistently track the vortex. Thus, we are limited to only the first 100 days of tracking the vortex with this method, after which the anomalous PV of the vortex was comparable to ambient noise. \n\n\nWe fit the anomalous PV at two levels: 1000 hPa and 500 hPa (Figure~\\ref{fig:cd_pv}), corresponding roughly to the vertical extent of the vortex. Below roughly 2000 hPa, the vortex is very weak and the anomalous PV is almost negligible. We find that the vortex is fairly uniform between the two layers, as the vortex is mostly equal in size and at the same position. The 1000 hPa anomalous PV decreases faster than the 500 hPa layer, and thus for all further analysis we report values from the vortex tracked using the 500 hPa anomalous PV.\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=1\\textwidth]{Figures\/panels5.png}\n    \\caption{Snapshots of the $40\\times$ solar methane abundance simulation showing CD fit (black), PV fit at 1 bar (red), and PV fit at 500 hPa (green) to the vortex, along with their respective centres ($\\times$). Methane column density is plotted in the background in blue, with the darker regions corresponding to lower CD (less vapour) with a lower limit of $-0.015$ kg\/m$^2$ and an upper limit of $0.4$ kg\/m$^2$ . The PV is shifted by about $1.5\\degree$ to the north east from the CD centre and decreases over time.}  \n    \\label{fig:cd_pv}\n\\end{figure*}\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{Figures\/pv_center_final.png}\n    \\caption{Temporal variation PV and anomalous PV\n    (both in units of m$^2$ K s$^{-1}$ kg$^{-1}$)\n    at 500 hPa inside the simulated GDS in our test cases using CD to track the vortex. In the dry case, only anomalous PV at 500 hPa was used to track the simulated GDS due to the lack of methane. } \n    \\label{fig:pv_evolution}\n\\end{figure}\n\n\\subsection{Drift Rate}\\label{Drift_rate}\n\\citet{LeBeauDowling_Neptune} investigated the equatorial drift of GDS-89 by using constant-vorticity gradient zonal wind profiles (Fig. \\ref{fig:zonal_wind}) and found that the value of the mid-latitude pseudo PV gradient, $Q_y$ is the primary influencer on meridionial drift. Using $Q_y=\\frac{1}{3}$, our experiments match the observed drift rate of the GDS-89 of $1.3\\degree$ per month \\citep{Sromovsky1993}.\n\nFigure \\ref{fig:drift_rate} shows the different cases investigated in this study and their latitudinal drift relative to the approximate observed drift rate of the GDS. Two lines per case are depicted, one for the PV at 500 hPa (dashed) and one for the methane column density (solid). Table \\ref{drift_rate_table} summarizes the average drift rate for each case investigated.\n\nCase 2 achieves the closest drift to the observed value of GDS-89, followed by Case 1. The  passive and dry cases (Cases 4 and 5 respectively) have the slowest drift. The $80\\times$ (Case 3) struggles to maintain dynamic stability throughout the simulation. We could not fit an ellipse around the vortex since it did not have a uniform shape and it dissipated after approximately 60 days. \n\n\\begin{table}\n\\centering\n\\begin{tabular}{ccc}\n\\hline\n\nCase    & CD Drift Rate [$\\degree$\/month] & PV Drift Rate [$\\degree$\/month]\\\\ \\hline\\hline\n1       & $0.82$   & $0.90$                                                   \\\\ \\hline\n2       & $1.36$ & $1.26$                                                     \\\\ \\hline\n3       & -        & -                                            \\\\ \\hline\n4       & $0.27$ & $0.31$                                                    \\\\ \\hline\n5       & - & $0.46$                                                   \\\\ \\hline\n\\end{tabular}\n\\caption{A summary of the approximate drift rate using either CD or PV at 500 hPa for each case. For case 3($80\\times$), we were unable to generate a fit. Case 5 is a dry model so PV alone was used to track the vortex.}\n\\label{drift_rate_table}\n\\end{table}\n\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{Figures\/drift_rate_final.png}\n    \\caption{Drift rate of the simulated GDS in our test cases. We used both methane column density (solid) and potential vorticity (dashed) at 500 hPa to fit the ellipse.  In the dry case, only PV at 500 hPa was used to track the simulated GDS due to the lack of methane. he dashed grey lines are the average meridional drift rate of the GDS-89 observed by {\\it Voyager 2} \\citep{Sromovsky1993}. }\n    \\label{fig:drift_rate}\n\\end{figure}\n\n\\subsection{Shape Oscillations}\\label{shape_oscillations}\nAlong with meridionial drift, time dependent oscillations in the shape of the vortex are of interest. Figure \\ref{fig:shape_oscillation} shows the inverted aspect ratio ($b\/a$) of the fitted ellipse for CD (Cases 1-4) and the PV at 1 bar (Case 5). In all cases, we correct for geometric effects by converting the extents to physical lengths using an equatorial radius of $24,760$ km and polar radius of $24,343$ km.  Clearly, these simulations exhibit complex dynamical variability in shape. The passive (Case 4) and dry (Case 5) cases have a lower inverted aspect ratio than the active cases (1-3) at around 0.3 as opposed to 0.4. Additionally, for the active cases, there does not appear to be a predictable frequency of oscillation and a Fourier power spectrum did not reveal any strong periodicity. The semi-major axis of the active cases are increasing without a corresponding increase in the minor axis resulting in a more elliptic vortex with time. Conversely, the passive and dry cases have the opposite occurring (more circular). \n\nThe GDS-89 was observed to increase in its semi-minor axis ($b$) at a rate of $3.8 \\times 10^{-5}$ h$^{-1}$ \\citep{Sromovsky1993} as it drifted northward, while the semi-major axis ($a$) was roughly constant, resulting a more circular vortex with time. This is in contradiction to our cloudy simulations where we see the vortex getting more elliptic by stretching along the zonal axis as it drifts northward while maintaining a nearly constant meridional height (Figure~\\ref{fig:cd_pv}). In the passive and dry case, there is very little change in the aspect ratio throughout the run, and they are much closer to the observed aspect ratio trend observed by {\\it Voyager 2}. \n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{Figures\/shape_oscillation_final.png}\n    \\caption{The inverted aspect ratio of the ellipse (semi-major divided by semi-minor) for the different test cases using methane column density to fit the simulated GDS. The solid black line is the average increase in the inverted aspect ratio of the GDS-89 observed by {\\it Voyager 2} \\citep{Sromovsky1993}. The period nature of the vortex (i.e. the `rolling motion') is evident in all cases, but at different time scales. }\n    \\label{fig:shape_oscillation}\n\\end{figure}\n\n\\subsection{Companion Clouds}\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{Figures\/comp_cloud.png}\n    \\caption{Model output at day 44. (a) shows the top-down view of methane column density (blue) and cloud (white). (b) shows the vertical wind with upwelling in red and downwelling in blue, with wind vectors in the comoving frame. (c) and (d) are longitudinal slices showing methane relative humidity, with brighter corresponding to saturated regions and darker being drier. Wind vectors show the local circulation within the vortex and the black contours are ice cloud densities. The location of the slices are shown in (a) in red. }\n    \\label{fig:comp_cloud}\n\\end{figure}\n\nFigure~\\ref{fig:comp_cloud} (a) shows a snapshot of the model output with the methane column density in blue and companion clouds in white. There are two distinct types of clouds that interact with the vortex: those that are interior (within the horizontal extents of the vortex) and that are exterior to the vortex (Figure~\\ref{fig:comp_cloud} a). The exterior clouds form a thin layer at roughly 1 bar (Fig~\\ref{fig:comp_cloud} c, d), while the interior clouds form much higher and are extended vertically.  \n\nThe interior clouds form at two discrete levels: one close to the tropopause at $\\sim100$ hPa and the other further down around $400-500$ hPa. The upper level cloud particles are small due to being in a thinner environment and do not reach the critical diameter of $500\\mu$m to precipitate. The humid environment where the clouds form (bright in Fig~\\ref{fig:comp_cloud} c, d) are retained throughout the vortex lifetime due to the thermal structure of the vortex. \n\nThe deeper cloud is denser and precipitates immediately. However, both clouds are long lived and last about 30 days after the initial adjustment period before dissipating. The particle sizes in both clouds, which are diagnosed from the cloud water content \\citep{Palotai2008}, are on the order of $300-500\\mu$m with the larger particles forming in the lower cloud. These sizes are comparable to cold cirrus clouds on Earth \\citep{Heymsfield2017}, which is a similar habit to the interior clouds in the GDS. \n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{Figures\/drift_rate_nds_final.png}\n    \\caption{Drift rate (blue) and aspect ratio (red) of NDS-2018 from model output. }\n    \\label{fig:NDS_drift}\n\\end{figure}\n\\subsection{NDS-2018}\n\nIn 2018, Hubble Space Telescope visible wavelength imaging revealed the presence of a dark spot in the northern hemisphere (labelled NDS-2018) \\citep{Simon2019}. OPAL observations showed that there was increase in active cloud formation in the regions for 1-2 years leading up to the development of the spot. The NDS-2018 is similar in size and shape to the GDS-89 \\citep{Simon2019}. Consequently, in order to constrain the dynamics of this feature, we run simulations with identical vortex initialization, domain, and resolution as the GDS cases. We run 100 day simulations of this feature using an active cloud microphysical model. The vortex was initialized at $32\\degree$ N latitude with a \n$40\\times$ solar [C\/H] fraction and an initial relative humidity of 95\\%. Fig. \\ref{fig:nds} shows select timesteps of this NDS simulation. \n\nDue to the symmetrical nature of the zonal wind profile used in this study (Fig \\ref{fig:zonal_wind}), equatorial drift was observed in the simulated NDS case similar to the simulated GDS cases (Fig. \\ref{fig:NDS_drift}). The average approximate drift rate found in this study for the NDS-2018 was roughly $3.2\\degree$ per month. In contrast to the surprisingly consistent latitudinal wobble observed in the southern hemisphere for the GDS cases, the NDS case has inconsistent oscillations in latitude. This simulation also saw a decrease in inverted aspect ratio throughout the 100 day simulation with an average value of around 0.4 consistent with the value of $b\/a=0.45$ observed by \\citet{Simon2019} for the NDS-2018.\n\n\nThe cloud formation in the NDS simulations were also similar to the GDS cases, with a set of interior and exterior clouds, although the interior clouds were not perfectly centered like the GDS simulations. These are likely a result of the much more irregular dynamics of the NDS vortex in our simulations. \n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=1\\textwidth]{Figures\/40x_shape_oscillation_nds.png}\n    \\caption{Select timesteps from a 100 day simulation of Neptune's NDS-2018 using the EPIC model with active microphysics for methane and $40\\times$ solar methane abundance. The blue is integrated column density of methane vapor with darker regions indicating areas of low density. The solid line is the fitted ellipse using dashed line, which is the vapor field contour.}\n    \\label{fig:nds}\n\\end{figure*}\n\n\\section{Discussion}\nLarge scale vortices that have been observed on Neptune exhibit surprising variability in terms of shape, drift, and lifetime. Consequently, they are some of the most dynamic features in the Solar System. Due to these variations, a variety of vortex characteristics can be examined at short timescales. Although we are constrained by limited observational data, recent developments in the accuracy of models using active microphysics can help us investigate the dynamics of these features. \n\n\\subsection{Methane deep abundance} \n\nThe various cases discussed here exhibit a wide range in both the meridionial drift and the dynamic time-dependent variations in shape of the GDS-89. \n\nWe tracked the vortex using both anomalous PV and methane column density (Figure~\\ref{fig:cd_pv}), and found that using the CD allows us to track the vortex for much longer due to the anomalous PV decreasing as the vortex moves through different shear environments. These cases differ significantly in the rate of latitudinal drift, the change in aspect ratio and the periodicity of the oscillations. Cases 1 and 2 ($20\\times$ and $40\\times$ solar methane abundance with active microphysics) are best matches for the drift rate (with Case 2 reproducing the drift almost perfectly), but do not show the observed decrease in ellipticity of the GDS-89. The passive and dry (Cases 4 and 5) are poor matches for the drift rate but are better at reproducing the observed slow change in the aspect ratio. Surprisingly, the $80\\times$ simulation failed to achieve a stable feature, and dissipates after 60 days. It should be noted that the high fidelity observations of the GDS started above $\\sim-26\\degree$ latitude \\citep{Sromovsky1993}, which is further north compared to where most of our simulations reach, and it is likely that we were probing an earlier part of the GDS-89's trajectory. A study of starting the vortex further north will need to be done to investigate this. Ultimately, the stability of Cases 1 and 2 and their consistency with {\\it Voyager 2} observations reinforces the measurements of a roughly $20-40\\times$ solar [C\/H] ratio on Neptune. \n\n\\subsection{Cloud microphysics}\n\nCloud microphysics has a much more drastic effect on the dynamics of the vortex compared to simply the addition of methane, as the passive case behaves similarly to the dry case. It is, however, much more difficult to trace the process(es) responsible for these effects. Due to the low density of the methane clouds, coupled with the low latent heat release of methane, it is unlikely to be just latent heating that drives this difference. \n\n\n\nClouds were present throughout the simulation and interacted strongly with the vortex. Similar to \\citet{Stratman2001}, we observed multi-layered clouds both within and without the vortex in our simulations (Figure~\\ref{fig:comp_cloud}). Their eastward cloud forms much higher than ours. However, this may be a function of vertical resolution and interpolation between the layers. In general, both results agree with the humid environment within the vortex and a region of dry air directly below. We are unable to reproduce the persistent poleward cloud. \n\\citet{Stratman2001} showed that vertical location of the vortex strongly influences the formation of clouds and indeed observed the poleward cloud only in certain configurations. The vertical extent and location of the spot in our simulations is held constant to reduce the number of free parameters, and we will investigate the effect of changing them in future simulations.\n\nThe deep cloud within the vortex formed snow and had cloud particles exceeded $300\\mu$m in diameter based on the assumption of hexagonal plate-like particles. Using {\\it Voyager 2} IRIS spectra, \\citet{Conrath1991} obtained particle sizes between $0.3-30\\mu$m, assuming Mie scattering by spherical particles. In our model, we assume ice and snow to be hexagonal plates (as determined for Earth clouds) resulting in much larger diameters than the equivalent spheres for the same particle masses. The equivalent radii for the spherical cloud particles in our simulations is about $50-70\\mu$m, for the same particle mass (i.e. the ``wetted radius\"). \\citet{Kinne1989} noted that using spherical ice crystals underestimates the albedo by up to $15\\%$ due to greater forward scattering, depending on the shape of the ice particle. Therefore, the discrepancy in cloud particle sizes on Neptune requires further study in the avenue of experimentally determining methane nucleation in a H\/He environment and scattering for non-spherical geometries. Both of these would further refine the microphysical parameters in our model to better represent gas-giant atmospheres. \n\n\n\\subsection{Optical depth}\n\nStrong methane absorption in Neptune's atmosphere largely correlates with a higher optical depth \\citep{Hueso2017}. In our simulations, we find that the vortex exists as an area of low methane vapor density compared to the surrounding atmosphere. This is due to the modified thermal structure of the atmosphere as a consequence of introducing the vortex (Figure~\\ref{fig:cd_pv}). All cases (except for Case 5 - which contains no methane vapor) reciprocate this phenomenon and decreased methane column density persist throughout the simulation. The reduced column density results in a decreased optical depth over the vortex compared to the ambient background atmosphere. We interpret this as being able to see deeper into the atmosphere, producing the ``darkness\" of the GDS-89. This is, indeed, an `apples-to-oranges' comparison, and applying a forward radiative transfer (RT) model to simulation outputs would lend additional insight into this discussion. We are working on an RT model that will address this in future studies. \n\n\n\\subsection{Applications to the NDS-2018}\n\nThe NDS-2018 simulation has surprising differences from our GDS test cases. There is much more irregularity in the equatorward drift of the vortex and a higher drift rate. The shape oscillation was similar to our wet GDS cases, with the vortex becoming more elliptic with time. \n\n\nThese differences could be due to the {\\it Voyager}-era zonal wind profile we use in our models. Recent OPAL observations have provided new zonal wind data and even shown evidence of weak vertical wind shear \\citep{Tollefson2018}. \\citet{Simon2019} suggest that a zonal wind gradient of $\\sim4\\times$ that of the \\citet{Sromovsky1993} fit is required to match the observed aspect ratio of the NDS-2018, or $du\/dy\\sim5.4\\times10^{-5}$ s$^{-1}$. The Q$_y=1\/3$ zonal wind profile we use has a value of $du\/dy\\sim 3\\times10^{-5}$ s$^{-1}$ around $30\\degree$ latitude. Testing different zonal wind gradients is not part of this study however, and will be introduced into future sims. Furthermore, without additional observations, it is difficult to verify the accuracy of the drift rate and shape oscillations we measured in the model. It is unknown whether the spot is still present because Hubble observations are limited. \n\n\n\\section{Conclusion}\n\nIn this study, we have analysed the effect of methane abundance and cloud formation on the dynamics of the vortex. While the addition of active microphysics improves the meridional drift rate of the vortex, there are other observations which are difficult to reproduce, such as the consistent periodicity of the shape oscillation or the increase in inverse aspect ratio. A further study of the parameter space that describes the spot is required, such as changing the initial vortex location and size (both vertically and horizontally). Additionally, \\citet{Tollefson2018} assumes a $\\times 2$ or an $\\times 4$ depletion of methane at the mid-latitudes and towards the poles and an increase in mixing ratio near the equator based on idealized equations. These recent observations would be beneficial in the study of the newer spots such as the NDS-2018, and indeed are likely required to explain their dynamics. \n\nThe active microphysics package implemented in this study has a clear effect on the dynamical motions of the vortex, such as meridional drift and shape oscillations. Using a $40\\times$ or $20\\times$ methane deep abundance, we achieve a stable simulation for over 120 days. We also see persistent bright companion cloud formation throughout the simulation. Additionally, the vortex exists as an area of low density which may be correlated to a decrease in optical depth. Finally, we found a possible scenario for the dynamical evolution of the NDS-2018 including its equatorial drift and the vortex becoming more elliptic with time. However, additional observations are needed to constrain these simulations. \n\n\\section*{Acknowledgements}\nThis research was supported in part by the NASA Solar System Workings Program grant NNX16A203G and the DPS Hartmann Student Travel Grant Program. N.H. thanks the Astronaut Scholarship Foundation for their support. We also thank Noah Nodolski for his help and the anonymous reviewer for their input.\n\n\\vspace{-1em}\n\\section*{Data Availability}\nThe data underlying this article will be shared on reasonable request to the corresponding author.\n\n\n\n\n\n\n\\vspace{-2em}\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nGame theory was originally developed in the field of economics to study strategic interactions amongst humans \\citep{neumann:1944ef,flood:1952aa}. \nThe ``agents\" who play against each other have a set of ``strategies\" to choose from.\nThe payoff which an agent gets depends on its own strategy and the strategy of the opponent.\nA player can decide which strategy to play against an opponent of a given strategy.\n\nIn evolutionary game theory players are born with fixed strategies instead, \\citep{maynard-smith:1982to}\nwhich are  considered to be inherited traits.\nAs usual, we assume a population game in which every player effectively plays against the average opponent.\nThe success of a strategy depends on the number of players of that strategy and also the number of players with other strategies.\nA classical example is the Lotka-Volterra equation \\citep{lotka:1910aa,volterra:1928aa,hofbauer:1998mm}.\nIf the number of wolves increases then the number of hares will decrease in turn leading to a decrease in the number of wolves.\nEvolutionary game dynamics studies the change in the frequencies of the strategies \\citep{nowak:2006bo}, which depends on mutation, selection and drift.\n\nA recurrent and obvious question asked in the study of games is which is the best strategy?\nAssuming an infinitely large population we can approach this question by the traditional replicator dynamics \\citep{hofbauer:1998mm}.\nThe frequency of a strategy will increase if its average payoff is greater than the average payoff of the whole population.\nThat is, if the individuals of a particular strategy are doing better on average than the individuals of other strategies then that strategy spreads.\nThe average payoff of a strategy is also dependent on the frequency of the strategy.\nFor finite populations one must resort to stochastic descriptions \\citep{ficici:2000aa,schreiber:2001aa,nowak:2004pw}.\nOne important quantity is the fixation probability.\nConsider two strategies $A$ and $B$ in a population of size $N$.\nLet the population be almost homogenous for $B$ with only a single $A$.\nIf there is no fitness difference amongst the strategies, i.e. selection is neutral, then the probability that the $A$ individual will take over the entire population is $1\/N$.\nIf this probability is greater than $1\/N$ we say that strategy $A$ is favoured by selection.\nWhen there are multiple strategies in the population, then a pair-wise comparison between the fixation probabilities of all the strategies will reveal which is the most abundant strategy \n\\citep{fudenberg:2006ee,hauert:2007aa,hauert:2008bb,van-segbroeck:2009mi,sigmund:2010aa}.\nThis analysis requires the assumption of low mutation rates.\n\nWhen mutations become more frequent then the concept of fixation itself is problematic and hence also that of fixation probability.\nIn such a case we resort to the average frequency of a strategy which is maintained at the mutation-selection balance.\nThis has been termed as the abundance of a strategy \\citep{antal:2009hc}.\n\nConsider $n$ strategies which are effectively neutral against each other.\nIn such a case the abundance of all the strategies in the stationary state will be just $1\/n$.\nUsually there are fitness differences between the strategies.\nIf the abundance of a strategy is greater than that of all the other strategies then we can say it is favoured under the effects of mutation, selection and drift.\nHence for $n$ strategies, the $k^{th}$ strategy will be favoured if the abundance of $k$ is greater than $1\/n$.\nCalculating the abundance of a strategy is a non-trivial exercise even when assuming weak selection.\n\\cite{antal:2009hc} have developed such an approach based on coalescence theory for the case of two-player games and $n$ strategies.\nUnder certain conditions and weak selection, one can calculate the most abundant strategy for arbitrary mutation rates even in structured populations \\citep{antal:2009aa,tarnita:2009df,tarnita:2009jx} and bimatrix games \\citep{ohtsuki:2010aa}.\n\nUsually two-player interactions are studied in evolutionary game theory.\nThe analysis of Antal et al. is also for two-player games.\nThe interactions which we usually use as examples in evolutionary game theory are in general multi-player interactions making the systems nonlinear \\citep{nowak:2010na}.\nA classical example where a certain minimum number of individuals are required to complete a task is group hunting.\n\\cite{stander:1992aa} studied cooperative hunting in lions.\nA typical hunting strategy is to approach the prey from at least three sides to cutoff  possible escape paths.\nThis hunting approach is impossible with only two hunters, i.e.\\ a two-player game theoretic approach would be insufficient to capture the dynamics.\nAlthough evolutionary dynamics of multi-player games has received growing interest in the recent years, the main focus has been the Public Goods Game and its variants \\citep{hauert:2002te,milinski:2006aa,rockenbach:2006aa,hauert:2007aa,santos:2008xr,pacheco:2009aa,souza:2009aa,veelen:2009ma,archetti:2011aa}.\nOnly few authors consider general evolutionary many person games  \n \\citep{hauert:2006fd,kurokawa:2009aa,gokhale:2010pn}.\nWe extend the approach developed by \\cite{antal:2009hc} for two-player games and multiple strategies to multi-player games.\nWe show that in the limit of weak selection it is possible to calculate analytical results for $n$ strategies and $d$ players for arbitrary mutation rates.\nFor a three-player game the mathematical analysis is described in detail.\nIt is followed by an example with simulations supporting the analytical result.\nLastly we discuss how the methodology can be extended for $d$-player games and argue that a general approach is possible, but tedious.\n\n\\section{Abundances in the stationary state for three-player games}\n\\label{model}\n\n\\cite{antal:2009hc} have developed an approach to find the abundances of $n$ strategies in a two-player game ($d=2$).\nFor a two-player game even with $n$ strategies, the payoff values can be represented in the usual payoff matrix form.\nThey can be represented as quantities with two indices, $a_{k,h}$.\nWe increase the complexity first by adding one more player ($d=3$).\nThis adds another index for the third player's strategy set, $a_{k,h,i}$.\nTo calculate the average change in the frequency of a strategy we thus need to take into account this payoff `tensor'.\n\nWe calculate the abundance of a strategy at the mutation-selection equilibrium.\nWe begin by checking if there is a change in the frequency of a strategy, say $k$ on average, due to selection.\nThe average change under weak selection is given by\n\\begin{eqnarray}\n\\label{replike}\n\\langle \\Delta x_k^{sel} \\rangle_\\delta = \\frac{\\delta}{N} \\left(\\sum_{h,i} a_{k,h,i}\\langle x_k x_h x_i\\rangle - \\sum_{h,i,j} a_{h,i,j}\\langle x_k x_h x_i x_j\\rangle \\right),\\nonumber \\\\\n\\end{eqnarray}\nwhere the angular brackets denote the average in the neutral stationary state.\nThe $\\delta$ (selection intensity) as a lower index on the left hand side, however, denotes that the average is obtained under (weak) selection.\nIf we pick three individuals in the neutral stationary state, then the probability of the first one to have strategy $k$, the next one $h$ and the last $i$, is given by the angular brackets in the first sum, $\\langle x_k x_h x_i\\rangle$.\nFurthermore, $a_{k,h,i}$ denotes the payoff values obtained by a strategy $k$ player when pitted against two other players of strategy $h$ and $i$.\nFor $n$ strategies the sums run from $1$ to $n$.\nThis equation is the special case of a $d=3$ player game.\nThe derivation for arbitrary $d$ is given in \\ref{eq1app}.\nThe above equation is similar to the replicator equation, which is also based on the difference between the average payoff of a strategy and the average payoff of the population, but as we will see below, here the averages on the right hand side also include mutations.\n\nTo incorporate mutations in the process, we write the total expected change due to mutation and selection as\n\\begin{eqnarray}\n\\label{xtot}\n\\Delta x ^{tot}_k = \\Delta x^{sel}_k (1-u)+ \\frac{u}{N}\\left(\\frac{1}{n} - x_k\\right).\n\\end{eqnarray}\nThe first term is the change in the frequency in the absence of mutation.\nIn presence of mutations, the second term shows that the frequency can increase by $1\/(nN)$ by random mutation and decrease by $x_k\/N$ due to random death.\nA mutation means that with a certain probability $u$, the strategy $k$ can mutate to any of the $n$ strategies.\n\nWe are interested in the abundance of a strategy in the stationary state.\nIn the stationary state, the average change in frequency is zero, $\\langle \\Delta x ^{tot}_k \\rangle_\\delta = 0$, as the mutations are balanced by selection.\nAveraging Eq.\\ \\ref{xtot} under weak selection thus gives us\n\\begin{eqnarray}\n\\label{abundeq}\n\\langle x_k \\rangle_\\delta = \\frac{1}{n} + N \\frac{1-u}{u} \\langle \\Delta x^{sel}_k \\rangle_\\delta.\n\\end{eqnarray}\nThis is our quantity of interest, the abundance of a strategy when the system has reached the stationary state.\nFor $d=2$ player games, this quantity is given by \\cite{antal:2009hc}.\nFor the abundance of a strategy to be greater than neutral, $\\langle x_k \\rangle_\\delta > \\frac{1}{n}$, the change in frequency in the stationary state due to selection must be greater than zero, $\\langle \\Delta x^{sel}_k \\rangle_\\delta >0$.\n\nThus, we need to resolve the right hand side of Equation \\ \\ref{replike}.\nConsider the first term in the brackets.\nIn the neutral stationary state the number of combinations in the sums reduces due to symmetry, e.g. $\\langle x_i x_j x_j \\rangle = \\langle x_j x_i x_j \\rangle = \\langle x_j x_j x_i \\rangle$.\nHence, we need to calculate only three different terms, $ \\langle x_1 x_1 x_1 \\rangle $, $ \\langle x_1 x_2 x_2 \\rangle $ and $ \\langle x_1 x_2 x_3 \\rangle $.\nAlso for $d$ player games, the terms in the sums are reduced.\nFor the second term in the brackets we need to calculate five different types of averages, $\\langle x_1 x_1 x_1 x_1\\rangle$, $\\langle x_1 x_2 x_2 x_2\\rangle$, $\\langle x_1 x_1 x_2 x_2\\rangle$, $\\langle x_1 x_1 x_2 x_3\\rangle$ and $\\langle x_1 x_2 x_3 x_4\\rangle$.\nThese averages are derived in the \\ref{averagesapp}.\nUsing an approach from coalescence theory, we derive $s_i$, the probability that $i$ individuals chosen from the stationary state all have the same strategy.\nHence $s_4$ is the probability that four individuals chosen in the stationary state all have the same strategy.\nIf there are in all $n$ strategies, then the probability that all have exactly strategy $1$ is $s_4\/n$.\nHence, $\\langle x_1 x_1 x_1 x_1 \\rangle = \\langle x_2 x_2 x_2 x_2 \\rangle = \\ldots = \\langle x_n x_n x_n x_n \\rangle = s_4\/n$.\nConversely, $\\bar{s}_i$ is the probability that if we choose $i$ individuals in the stationary state, each has a unique strategy.\nKnowing these averages helps us resolve Eq.\\ \\eqref{replike},\n\\begin{eqnarray}\n\\langle \\Delta x_k^{sel} \\rangle_\\delta\n= \\frac{\\delta \\mu (L_k + M_k \\mu + H_k \\mu^2)}{N n (1+\\mu) (2+\\mu) (3+\\mu)} \n\\end{eqnarray}\nwhere $\\mu = N u$ and $L_k$, $M_k$ and $H_k$ are functions consisting only of the number of  strategies $n$ and the payoff values $a_{k,h,i}$ (see \\ref{averagesapp}).\nUsing this and evaluating Eq.\\ \\eqref{abundeq} gives us the abundance of the $k^{th}$ strategy.\n\\begin{eqnarray}\n\\label{finalres}\n\\langle x_k \\rangle_\\delta = \\frac{1}{n}\\left[ 1 + \\frac{ \\delta (N- \\mu) (L_k + M_k \\mu + H_k \\mu^2) }{ (1+ \\mu) (2+ \\mu) (3+ \\mu)}\\right].\n\\end{eqnarray}\nThis expression is valid for large population sizes, $N\\delta \\ll 1$ and any $\\mu =N u$.\nSince we have $0\\leq u \\leq 1$, $\\mu$ is bounded by $0\\leq \\mu \\leq N$.\n\n\nWe arrive at the result with an implicit assumption that there are at least four strategies.\nFor $n \\leq d$, each player cannot have a unique strategy and hence we need to set the corresponding terms to zero (see \\ref{averagesapp}).\nIf there are less than $n=4$ strategies then $\\bar{s}_4$ vanishes.\nThis does not affect our general result as the affected terms in $L_k$, $M_k$ and $H_k$ simply vanish.\n\n\n\\section{An example for three-player games with three strategies}\n\\label{example}\n\nTo illustrate the analytical approach we explore an evolutionary three-player game with three strategies $A$, $B$ and $C$.\nLet our focal individual play strategy $A$.\nThe other two players can play any of the three strategies.\nThis can lead to a potential complication.\nConsider the combinations $AAB$ or $ABA$.\nIf the order of players does not matter, then both these configurations give the same payoffs but if they do matter then we need to consider them separately.\nHere we assume random matching, and hence the order drops out (e.g. $AAB$ and $ABA$ are equally likely).\nWe consider an arbitrary game as denoted in Table \\ref{paytab}.\n\nWe need to calculate the average change in the frequency of strategy $A$ due to selection, i.e. Eq.\\ \\eqref{replike}.\nWe denote the co-efficients of the averages in the first sum by $\\alpha_1$, $\\alpha_2$ and $\\alpha_3$.\nHence for example, $\\alpha_3 = a_{A,B,C} + a_{A,C,B}$.\nSimilarly for the second sum we have $\\beta_1$ to $\\beta_4$ (Note that $\\beta_1 = \\alpha_1 = a_{A,A,A}$).\nThus we have,\n\\begin{eqnarray}\n\\sum_{h,i} a_{A,h,i}  \\langle x_A x_h x_i\\rangle &=& \\alpha_1 \\langle x_A x_A x_A \\rangle + \\alpha_2  \\langle x_A x_B x_B \\rangle \\nonumber \\\\\n&&+ \\alpha_3 \\langle x_A x_B x_C \\rangle\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\sum_{h,i,j} a_{h,i,j}\\langle x_A x_h x_i x_j\\rangle  &=& \\beta_1 \\langle x_A x_A x_A x_A\\rangle + \\beta_2 \\langle x_A x_B x_B x_B\\rangle \\nonumber \\\\\n&&+ \\beta_3 \\langle x_A x_A x_B x_B\\rangle + \\beta_4 \\langle x_A x_A x_B x_C\\rangle.\\nonumber \\\\\n\\end{eqnarray}\nNote that the term $\\langle x_A x_B x_C x_D\\rangle$ which would appear with a factor $\\beta_5$, does not appear, as we have only three strategies and thus $\\bar{s}_4 = 0$.\nThis also alters the definition of  $\\langle x_A x_A x_B x_C\\rangle$ and $\\langle x_A x_A x_B x_B\\rangle$ (see Figure \\ref{fig:1}, all terms dependent on $\\bar{s}_4$ are affected).\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figure_1.pdf}\n\\caption{The average change in the frequency of strategy $k$ due to selection, $\\langle \\Delta x_k^{sel} \\rangle_\\delta$ for a three-player game.\nNotice first the similarity to the replicator equation where also we look at how a strategy is faring compared to the population.\nThe first term in the bracket is analogous to the average fitness of strategy $k$.\nIf we pick three individuals in the stationary state, then the probability that the first one has strategy $k$, second $h$ and the third $i$ is given by $\\langle x_k x_h x_i\\rangle$  (dashed box).\nEven for $n$ strategies there are only three possible combinations, either all can have the same strategy, a pair has the same strategy or all three have different strategies.\nThese probabilities were calculated by \\cite{antal:2009hc}.\nThe $s_i$'s appearing in the averages are the probabilities that if we choose $i$ individuals from the stationary distribution then they all have the same strategy.\nThe second term in the bracket is analogous to the average fitness of the population in the stationary state.\nFor this we need to pick four individuals and look for all the different combinations (solid box).\nFor $n$ strategies, five combinations can explain all the different configurations.\nThese range from all the individuals having the same strategy $\\langle x_1 x_1 x_1 x_1 \\rangle$ to all having a different strategy $\\langle x_1 x_2 x_3 x_4 \\rangle$ (\\ref{averagesapp}).\nFor the latter, we calculate $\\bar{s}_i$, the probability that we choose $i$  individuals from the stationary distribution and each of them has a unique strategy.\nFor a general $d$-player game we need to pick $d$ individuals for the first term and $d+1$ for the second.}\n\\label{fig:1}\n\\end{figure}\n\n\n\nWe know the form of $L_k$, $M_k$ and $H_k$ from \\ref{averagesapp} as,\n\\begin{eqnarray}\nL_k  &=& \\tfrac{1}{n} \\left[2 \\alpha_1 (n-1) + 3 \\alpha_2 - 2 \\beta_2 - \\beta_3\\right] \\\\\nM_k &=& \\tfrac{1}{n^2} \\left[\\left(3 n-3\\right) \\alpha_1 +\\left(n+3\\right) \\alpha_2 +3 \\alpha_3 - 3 \\beta_2 - 2 \\beta_3 - \\beta_4\\right] \\nonumber \\\\\n\\\\\nH_k &=& \\tfrac{1}{n^3} \\left[n (\\alpha_1 + \\alpha_2 + \\alpha_3) - (\\beta_1 + \\beta_2  + \\beta_3  + \\beta_4  + \\beta_5)\\right]\n\\end{eqnarray}\nWith $L_k$, $M_k$ and $H_k$ as above, Eq.\\ \\eqref{finalres} for $n=3$ reduces to,\n\\begin{eqnarray}\n\\langle x_A \\rangle_\\delta = \\frac{1}{3}\\left[ 1 + \\frac{\\delta (N- \\mu) (L_k + M_k \\mu + H_k \\mu^2) }{ (1+ \\mu) (2+ \\mu) (3+ \\mu)}\\right].\n\\end{eqnarray}\nThis gives us the abundance of strategy $A$ at the mutation selection equilibrium.\nRepeating the same procedure for strategies $B$ and $C$ gives the analytical lines in \nFigure \\ref{fig:2}.\nAlthough the analytical solutions are valid for large population sizes only, we still see a good agreement between the simulation and theory results, even for a population size as small as $30$.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figure_2.pdf}\n\\caption{\nFor a three-player game with three strategies ($d=3;n=3$) we plot the average abundances of the three strategies as a function of the mutation probability $u$.\nThe payoff table from Table \\ref{paytab} is used.\nThe lines are the solutions of Eq.\\ \\eqref{abundeq} and the symbols are the simulation results for the three strategies.\nAlthough the calculations are valid for large populations we see a good agreement even for a population size of  $N=30$\n(selection intensity $\\delta=0.003$, simulation points are obtained \naveraging over $20000$ independent runs, each over $2 \\times 10^6$ time steps after a transient phase of $N$ time steps).\n}\n\\label{fig:2}\n\\end{figure}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ccccccc}\n\\hline\\hline\n\\hspace{0.2cm}\n\\begin{minipage}[c]{2cm}\n\\begin{center}\n\\vspace{0.1cm}\nWeights\n \\\\ \n( Total 9 )\n\\vspace{.2cm}\n\\end{center}\n\\end{minipage} & 1\t& 2 & 2 & 1 & 2 & 1\t\\\\\n\\hline\n & AA & AB & AC & BB & BC & CC\t\\\\\n \\hline\n A \t& 2 & 2  & 3 & 1 & 3 & 4\t\\\\\n B \t& 2 & 1 & 2 & 3\t& 0 & 2\\\\\n C \t& 2 & 12 & 2 & 0 & 1 & 3\t\\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{An example payoff table for $d = 3$ and $n=3$.\nConsider a three-player game with three strategies $A$, $B$ and $C$.\nThe strategy of the focal individual is in the column on the left.\nFor example the payoff received by a $C$ individual when playing in a configuration of $CAB$ is  $12$.\nFrom the focal individual's point of view there are two ways of this configuration $CAB$ and $CBA$ as it is twice as likely as compared to e.g. $CAA$.\nHence we weight that payoff value by $2$ when calculating the average payoff of strategy $C$.}\n\\label{paytab}\n\\end{center}\n\\end{table}\n\n\\section{Abundances in $\\mathbf{d>3}$ player games.}\n\nWe can repeat the whole procedure for $d=4$ player games with $n$ strategies.\nThe formula for the abundance remains the same, Eq.\\ \\eqref{abundeq}, but the average change due to selection, Eq.\\ \\eqref{replike}, becomes more complicated.\nWe need to add an index in the sums,\n\\begin{eqnarray}\n\\langle \\Delta x_k^{sel} \\rangle_\\delta &=& \\frac{\\delta}{N} \\Big(\\sum_{l,m,n} a_{k,l,m,n}\\langle x_k x_l x_m x_n\\rangle \\nonumber \\\\\n&&- \\sum_{l,m,n,o} a_{l,m,n,o} \\langle x_k x_l x_m x_n x_o\\rangle \\Big)\n\\end{eqnarray}\nThe first term is comparatively simple as we already know all the different ways of picking four individuals.\nFor the second term we need to know the different possible combinations of strategies when picking five individuals from the neutral stationary state.\n\nFor $d$ players and $n$ strategies we can construct an expression analogous to Eq.\\ \\ref{replike}.\nConsider for example the strategies played by $d$ individuals denoted by, $r_1, r_2, r_3 \\ldots r_d$.\nNote that each of these can be a strategy from the strategy set $1,2,3 \\ldots n$.\nLet $k$ be our strategy of interest.\nThen the expression for the change of strategy $k$ due to selection is given by,\n\\begin{eqnarray}\n\\label{dplayers}\n\\langle \\Delta x_{k}^{sel} \\rangle_\\delta &=& \\frac{\\delta}{N}  \\Bigg( \\sum_{r_2,\\ldots r_d} a_{k,r_2,\\ldots r_d } \\langle x_k x_{r_2} x_{r_3} \\ldots x_{r_d} \\rangle  \\nonumber \\\\\n\\!\\!\\!\\! &&- \\sum_{r_2,\\ldots r_{d+1}} a_{r_2,\\ldots r_{d+1} } \\langle x_k x_{r_2} x_{r_3} \\ldots x_{r_{d+1}} \\rangle \\Bigg) \\nonumber \\\\\n\\end{eqnarray}\nwhere the sums range as usual from $1$ to $n$ (\\ref{eq1app}).\nSolving this and plugging it in Eq.\\ \\eqref{abundeq} gives the generalized expression for the abundance of strategy $k$ for an $n$ strategy, $d$-player game.\nWe see that in the first sum the averages are for choosing $d$ players but for the second it is $d+1$.\nHence we need to calculate the probabilities of the form $s_{d+1}$, but $s_{d+1}$ depends on $s_d$.\nThus we have to solve the expression recursively e.g.\nfor $d=6$, we will need to pick $7$ players at most and we must solve the expression for $d = 2,3,4,5,6$ before ($d=2$ has been solved by \\cite{antal:2009hc} and $d=3$ in this paper).\nAs $d$ increases calculating $s_{d+1}$ is not enough and we will also need to calculate terms such as $\\bar{s}_{d+1}$ which is already the case for $d=3$.\n\n\\section{Special case: Two strategies, $\\mathbf{n=2}$}\n\nGames with two strategies have been very well studied.\nIn two-player games with two strategies, one strategy can replace another with a higher probability if the sum of its payoff values is greater than the sum of the payoff values of the other strategy.\nThis is valid under small mutation rates for deterministic evolutionary dynamics \\citep{kandori:1993aa}.\nThe result also holds for for different dynamical regimes under specific limits of selection intensity and mutation rates \\citep{fudenberg:1992bv,nowak:2004pw,antal:2009th}.\nRecently it has been shown that this result can be generalized for $d$-player games with two strategies \\citep{kurokawa:2009aa,gokhale:2010pn}.\n\nHence the condition which we find for $d$-player games should be identical to $L_k > 0$ derived in this paper for $d$ players.\nWe check this for $d=3$,\n\\begin{eqnarray}\nL_k  &=& \\tfrac{1}{2} \\left[2 \\alpha_1 (2-1) + 3 \\alpha_2 - 2 \\beta_2 - \\beta_3 \\right]\n\\\\\n\\nonumber \n&=&\\tfrac{1}{2}  [ 2 a_{1,1,1} + 3 ( a_{1,1,2} + a_{1,2,1} + a_{1,2,2} )\\nonumber \\\\ \n&& - 2 ( a_{1,1,2} + a_{1,2,1} + a_{2,1,1} + a_{2,2,2} ) - a_{1,2,2} - a_{2,1,2} - a_{2,2,1} ] \\nonumber \\\\\n\\end{eqnarray}\nThus $L_k > 0$ is equivalent to,\n\\begin{eqnarray}\n2 a_{1,1,1} +&&   a_{1,1,2} + a_{1,2,1} +  2 a_{1,2,2} \\nonumber \\\\\n&&> 2 a_{2,1,1} +   a_{2,1,2} + a_{2,2,1} +  2 a_{2,2,2} \\nonumber \\\\\n\\end{eqnarray}\nIf we assume that the order of players does not matter then we have $a_{1,1,2} = a_{1,2,1}$ and $a_{2,1,2} = a_{2,2,1}$.\nThis yields\n\\begin{eqnarray}\na_{1,1,1} +   a_{1,1,2} +  a_{1,2,2} >  a_{2,1,1} +   a_{2,1,2} +  a_{2,2,2},\n\\end{eqnarray}\nwhich is exactly the condition that has been obtained previously using different methods and different notation by \\cite{kurokawa:2009aa} and \\citep{gokhale:2010pn}.\n\\section{Application to a task allocation problem}\nTo demonstrate the power of the approach we are motivated by \\cite{wahl:2002aa} who studies the problem of task allocation and of the evolution of division of labour.\n\\cite{wahl:2002aa} studied a two-player game between task $1$ specialists ($T_1$), task $2$ specialists ($T_2$) and generalists. \nInstead, we have $T_1$, $T_2$ and freeloaders $F$.\nWe can think of this problem  in context of bacteria that need two types of enzymes to obtain resources from the environment.\nOne strain produces one type of enzyme at a cost $c_1$ and \nanother strain produces the second type of enzyme at a cost of $c_2$.\nWe also have the freeloading strain which does not produce any enzyme but can get resources by the help of the other two strains.\nThe benefit of getting \nthe resources is given by $b$.\nWe have the condition that the total cost is less than the benefit accrued i.e. $b> c_1 + c_2$.\nFurther assume that our contenders are conservative in the enzyme production.\nSensing who they are pitted against, the strains share the costs of producing the enzyme.\nThus a two-player payoff matrix for such a setting can be written down as in Table \\ref{2plpaytabex}.\nIt is hard to imagine though that the bacteria interact only in a pair-wise fashion.\nAlthough it is hard to judge how many players are interacting, we can at least increase the complexity by one more player and study what effect this has on the abundances of the strains.\nTherefore, we study the system\nin a three-player setting.\nIn this case the payoff table will look like \\ref{3plpaytabex}.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ccccccc}\n\\hline\\hline\n & $T_1$ & $T_2$ & $F$ \t\\\\\n \\hline\n$T_1$ & $\\frac{-c_1}{2}$ & $b-c_1$  & $-c_1$ \t\\\\\n$T_2$& $b-c_2$ & $\\frac{-c_2}{2}$ & $-c_2$ \\\\\n$F$ \t& 0 & 0 & 0 \\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{\nPayoff table for a two-player game with three strategies, $T_1$ i.e. specialising in task $1$ or $T_2$ i.e. specialising in task $2$.\n$T_1$ produces\nenzyme $1$ and \n$T_2$ produces \nenzyme $2$.\nWhen both the enzymes are present then a benefit $b$ is obtained.\nThe cost of producing enzyme $1$ is $c_1$ and the cost to produce enzyme $2$ is $c_2$.\nIf the partner in the game is of the same strategy then the cost is shared.\nThe freeloading strategy $F$ does not pay any cost and so the benefits of the resource are unavailable to it.}\n\\label{2plpaytabex}\n\\end{center}\n\\end{table}\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ccccccc}\n\\hline\\hline\n\\hspace{0.2cm}\n\\begin{minipage}[c]{2cm}\n\\begin{center}\n\\vspace{0.1cm}\nWeights\n \\\\ \n( Total 9 )\n\\vspace{.2cm}\n\\end{center}\n\\end{minipage} & 1\t& 2 & 2 & 1 & 2 & 1\t\\\\\n\\hline\n & $T_1 T_1$ & $T_1 T_2$ & $T_1 F$ & $T_2 T_2$ & $T_2 F$ & $F F$\t\\\\\n \\hline\n $T_1$ \t& $\\frac{-c_1}{3}$ & $b-\\frac{c_1}{2}$  & $\\frac{-c_1}{2}$ & $b-c_1$ & $b-c_1$ & $-c_1$\t\\\\\n $T_2$ \t& $b-c_2$ & $b-\\frac{c_2}{2}$ & $b-c_2$ & $\\frac{-c_2}{3}$\t& $\\frac{-c_2}{2}$ & $-c_2$\\\\\n $F$ \t& 0 & b & 0 & 0 & 0 & 0\t\\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{\nPayoff table for the same game as discussed in Table \\ref{2plpaytabex}.\n$T_1$ and $T_2$ refer to specialising in task $1$ and $2$ namely producing enzyme $1$ and $2$.\nNote the costs can also be shared if at least one of the game partners is of the same strategy.\nIn this case the freeloaders can get the benefit when the other two players produce both enzymes.\n}\n\\label{3plpaytabex}\n\\end{center}\n\\end{table}\n\nWe calculate the abundances of the strains in these different settings, cf.\\ Figure \\ref{fig:3}. \nEven when there are almost no mutations there is a quantitative difference between the average frequencies of the strains.\nFor a higher mutation probability the difference is also qualitative.\nWhile the freeloaders never pay a cost they have the highest abundance in the two-player setting for any mutation probability.\nFor the same reasoning but in three-player games we see that the abundance of the freeloaders falls below that of $T_2$ for a certain range of mutation probability.\n\n\n\\begin{figure*}\n\\includegraphics[width=2.0\\columnwidth]{figure_3.pdf}\n\\caption{\nThe strains of a bacteria $T_1$ and $T_2$ produce the enzymes $1$ and $2$ which when present together provide a benefit $b$.\nWhen more than one individual of the same strain is present then the production costs for the enzymes, $c_1$ and $c_2$ are shared.\nA third strain $F$ does not produce any enzyme and thus avoids the costs and cannot obtain the benefit on its own.\nWe can analyse the interactions by assuming them to be pair-wise (two-player game, Table \\ref{2plpaytabex}) or in triplets (three-player game, \\ref{3plpaytabex}).\nWe notice that the abundances of the three strains are qualitatively and quantitatively different in the two settings even though the underlying rules defining the interactions are the same.\nUnder neutrality the abundance of all the strains would be given by the dashed line.\nThe full lines are the analytical results obtained by solving Eq.\\ \\eqref{abundeq}, assuming $b=1.0$, $c_1 = 0.6$, $c_2 = 0.2$ and a population size of $N=30$ with selection intensity $\\delta =0.003$.\n}\n\\label{fig:3}\n\\end{figure*}\n\n\n\\section{Discussion}\n\nPublic goods games are often used as examples of multi-player games.\nIn the beginning there were the cooperators and defectors.\nThen came the punishers and then the loners \\citep{hauert:2002te,szabo:2002te}.\nNow we talk about second order punishers, pool and peer punishers \\citep{sigmund:2010aa} and more.\nStudying these systems for small mutation rates and arbitrary selection intensity is almost becoming standard \\citep{fudenberg:2006ee,hauert:2007aa,hauert:2008bb,van-segbroeck:2009mi,sigmund:2010aa}.\nIn the limit of weak selection our method allows to find out which strategy is most abundant for arbitrary mutation rates.\n\nYet, another important aspect of most social dilemmas and many other biological examples is that they involve multiple players \\citep{stander:1992aa,broom:2003aa,milinski:2008lr,levin:2009aa}.\n\\cite{antal:2009aa,antal:2009hc} have made use of the coalescence approach to characterize the mutation process under neutrality and then apply it under weak selection to two-player games with $n$ strategies ($n \\times n$).\nHere we extend the approach to $d$-player games with $n$ strategies.\n\n\nWe give an example for an $n \\times n \\times n$ game and derive the analogous expressions for abundances of the strategies for arbitrary mutation rates.\nWhen we increase the number of players to $d$, the payoff matrix becomes a $d$ dimensional object.\nWe run into the problem of whether the order of players matters or not.\nEither way this does not influence our results but notation-wise it is easier if the order of players does not matter.\nAdding a new player adds a new index to the payoff values.\nFor calculating the abundance we need to assess Eq.\\ \\eqref{dplayers}.\nFor solving the two sums in Eq.\\ \\eqref{dplayers} we need to know the different combinations of choosing $d$ players and $d+1$ players from the neutral coalescent stationary state.\n\nTo illustrate the complexity of the situation take for example $s_4$.\nThis is the probability that four chosen individuals have the same strategy.\nIn \\ref{coalescentapp} we have shown that deriving $s_4$ depends on $s_3$ which depends on $s_2$ in turn.\nHence in general to derive $s_{d+1}$, we need to know $s_{d}, s_{d-1}, s_{d-2} \\ldots , s_2$.\nIn addition, for general $d$-player games we need quantities such as $\\bar{s}_{d+1}$, probability that $d+1$ individuals chosen in the stationary state all have different strategies.\nIn the absence of mutations such probabilities as $\\bar{s}_{d+1}$ vanish and we are left with only $s_{d+1}$, as mentioned in \\cite{ohtsuki:2010bb}.\nIf $n<d$ then $\\bar{s}_{d+1}$ is zero and hence the terms dependent on it need to be recalculated.\nAfter recalculation the terms which are affected either vanish or are automatically adjusted such that the result can be written again in the form for $L_k$'s, $M_k$'s and $H_k$'s.\nHowever, for a $d$ player game, $d-2$ intermediate terms such as $M_k$ appear.\n\nFor two strategies, $L_k$ reduces to the general condition derived in \\citep{kurokawa:2009aa,gokhale:2010pn} and again holds for arbitrary $d$.\nFor $H_k$ for any number of strategies we conjecture that it will always be of the form $\\frac{1}{n^{d}}\\left[n \\left(\\sum_{r_2,\\ldots r_d} a_{k,r_2,\\ldots r_d } \\right)-\\left(\\sum_{r_2,\\ldots r_{d+1}} a_{r_2,\\ldots r_{d+1} } \\right)\\right]$ from Eq.\\ \\eqref{dplayers}.\nAddressing a generalization for $d$-player games is not a fundamental problem of the approach but requires a tedious \nrecursive effort and poses a notational challenge.\nAt the coalescence level the problem rests on permutations and combinations.\n\nArbitrary mutation rates can be interpreted in different ways.\nIn the social learning context \\citep{traulsen:2009aa} it can be thought of as the exploration rate where the players experiment with different strategies.\nA recent behavioral experiment  suggests that humans change their strategies with a probability much higher than usually considered in theoretical models \\citep{traulsen:2010pn}.\nAnalytical results for such high mutation rates can now be provided utilising the approach outlined herein.\nSmall mutation rates are most relevant in population genetic contexts where the strategies can be thought of as alleles.\nWhile most people think of  evolutionary game theory as a phenotypic approach, \none can as well consider evolutionary games on the level of genes \\citep{hofbauer:1982mm,sigmund:1987aa,rowe:1987aa,rowe:1988aa,cressman:1992aa,hofbauer:1984mm,hofbauer:1998mm}.\nA long standing problem in theoretical biology is to explain the maintenance of polymorphisms.\nSo far, the corresponding models typically assumed small mutation rates, but our approach provides a way to calculate the abundances for arbitrary mutation rates.\nHowever, our aim is not to calculate abundances, but to depict a way to derive them.\nA two-player interaction can capture the effect of one allele on another, but the epistatic interaction of many loci\ncalls for a theory of many-player interactions.\n\n\nMaking use of our approach we can precisely determine the composition of a population with any finite number of different types under weak selection for arbitrary mutation rates.\nAnother convenient way of finding the strategy which performs the best is the limit of small mutation rates.\nFor small mutation rates, the system spends most of its time in a monomorphic state.\nWe can approximate the system by just looking at fixation probabilities of the different types.\nOur approach illustrates that the interaction of $d$ players is significantly more complex than the usual two-player games.\nGeneral multi-player games pose exciting challenges way beyond the usual intricacies of public goods games.\\\\\n\n\\textbf{Acknowledgements}. The authors would like to thank Bin Wu for helpful discussions and suggestions throughout the preparation of the manuscript.\nThe suggestions of two anonymous referees are highly appreciated.\nC.S.G. and A.T. acknowledge support from the Emmy-Noether program of the Deutsche Forschungsgemeinschaft and from the Max-Planck Society.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe large sieve was an idea originated by J. V. Linnik \\cite{JVL1} in 1941 while studying the distribution of quadratic non-residues.  Refinements and extensions in various directions of this idea were made by many \\cites{Ba1, Ba2, PrRa, SBLZ, SBLZ3, BD1, Da, DH1, DRHB, Cla, B&Y, PXG, JVL1, Lem, HM2, MVa, Wol, Zha, Zha2, Zha3, Zha4}.  Large sieve results for Dirichlet characters with a fixed order\nare particularly useful in analytic number theory. We refer the readers to\n\\cite{Elliott}, Section 7, for some early large sieve-type results\non general $r$-th order characters.\n Let $(a_n)_{n\\in \\mathbb{N}}$ be an arbitrary sequence of complex numbers, D. R. Heath-Brown's quadratic large sieve \\cite[Theorem 1]{DRHB} states\n that for any $\\varepsilon>0$,\n\\begin{equation}\n\\label{realfinal}\n   \\sideset{}{^*}\\sum_{m \\leq M} \\left| \\sideset{}{^*}\\sum_{n \\leq N} a_n \\Big (\\frac {n}{m} \\Big ) \\right|^2 \\ll_{\\varepsilon} (MN)^{\\varepsilon}(M+N)\n   \\sideset{}{^*}\\sum_{n \\leq N} |a_n|^2,\n\\end{equation}\n   where the asterisks indicate that $m,n$ run over positive odd square-free\n   integers and $(\\frac {\\cdot}{m})$ is the Jacobi symbol. \\newline\n\n   Similar to \\eqref{realfinal}, Heath-Brown also established the following large sieve inequality involving the cubic symbols \\cite[Theorem\n   2]{DRHB1}:\n\\begin{equation}\n\\label{eq:HBcubic}\n \\sideset{}{^*}\\sum_{\\substack{m \\in \\ensuremath{\\mathbb Z}[\\omega] \\\\\\mathcal{N}(m) \\leq M}} \\left| \\ \\sideset{}{^*}\\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[\\omega] \\\\\\mathcal{N}(n) \\leq N}} a_n \\leg{n}{m}_3 \\right|^2\n \\ll \\left( M + N + (MN)^{2\/3} \\right)(MN)^{\\varepsilon} \\sum_{\\mathcal{N}(n) \\leq N} |a_n|^2,\n\\end{equation}\nwhere the asterisks indicate that $m,n$ run over square-free elements\nof $\\ensuremath{\\mathbb Z}[\\omega], \\omega=\\exp(2 \\pi i\/3)$ that are congruent to $1$ modulo 3 and $(\\frac {\\cdot}{m})_3$ is the cubic residue symbol.  Moreover, here and after, we use $\\mathcal{N}(m)$ to denote the norm of $m$. \\newline\n\n   Using \\eqref{eq:HBcubic}, S. Baier and M. P. Young \\cite[Theorem 1.4]{B&Y} proved the following large sieve inequality for cubic Dirichlet characters:\n\\begin{equation*}\n\\begin{split}\n& \\sum\\limits_{\\substack{Q<q\\le 2Q}} \\\n\\sideset{}{^\\star}\\sum\\limits_{\\substack{\\chi \\bmod q\\\\ \\chi^3=\\chi_0}}\n\\left| \\ \\sideset{}{^*}\\sum\\limits_{\\substack{M<m\\le 2M}} a_m\n\\chi(m)\\right|^2\\\\\n&\\ll\n(QM)^{\\varepsilon}\\min\\left\\{Q^{5\/3}+M,Q^{4\/3}+Q^{1\/2}M,Q^{11\/9}+Q^{2\/3}M,\nQ+Q^{1\/3}M^{5\/3}+M^{12\/5}\\right\\}\n\\sideset{}{^*}\\sum\\limits_{\\substack{M<m\\le 2M}}\\left| a_m \\right|^2,\n\\end{split}\n\\end{equation*}\nwhere the star on the sum over $\\chi$ restricts the sum to\nprimitive characters and the asterisks attached to the sum over $m$\nindicates that $m$ runs over square-free integers. \\newline\n\nIt is our goal in this paper to prove a large sieve inequality for quartic\nDirichlet characters.  First we prove the following theorem involving the quartic symbols.\n\\begin{theorem} \\label{mainthm}\nLet $M,N$ be positive integers, and let $(a_n)_{n\\in \\mathbb{N}}$ be an arbitrary\nsequence of complex numbers, where $n$ runs over $\\ensuremath{\\mathbb Z}[i]$. Then we\nhave\n\\begin{equation} \\label{eq:quartic}\n \\sideset{}{^*}\\sum_{\\substack{m \\in \\ensuremath{\\mathbb Z}[i] \\\\\\mathcal{N}(m) \\leq M}} \\left| \\ \\sideset{}{^*}\\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i] \\\\\\mathcal{N}(n) \\leq N}} a_n \\leg{n}{m}_4 \\right|^2\n \\ll_{\\varepsilon} \\left( M + N + (MN)^{3\/4} \\right)(MN)^{\\varepsilon} \\sum_{\\mathcal{N}(n) \\leq N} |a_n|^2,\n\\end{equation}\n   for any $\\varepsilon > 0$,where the asterisks indicate that $m$ and $n$ run over square-free elements of $\\ensuremath{\\mathbb Z}[i]$ that are congruent to $1$ modulo $(1+i)^3$ and $(\\frac\n{\\cdot}{m})_4$ is the quartic residue symbol.\n\\end{theorem}\n\n   Next we shall establish the following large sieve inequality for quartic Dirichlet characters.\n\\begin{theorem}\n\\label{quarticlargesieve} Let $(a_m)_{m\\in \\mathbb{N}}$ be an\narbitrary sequence of complex numbers. Then\n\\begin{equation} \\label{final}\n\\begin{split}\n&  \\sum\\limits_{\\substack{Q<q\\le 2Q}}  \\\n\\sideset{}{^\\star}\\sum\\limits_{\\substack{\\chi \\bmod q\\\\ \\chi^4=\\chi_0,\n\\chi^2 \\neq \\chi_0}} \\left| \\ \\sideset{}{^*}\\sum\\limits_{\\substack{M<m\\le 2M}}\na_m \\chi(m)\\right|^2\\\\ &\\ll\n(QM)^{\\varepsilon}\\min\\left\\{Q^{7\/4}+M,Q^{11\/8}+Q^{1\/2}M, Q^{5\/4}+Q^{2\/3}M,\nQ+Q^{1\/2}M+M^{17\/7}\\right\\} \\sideset{}{^*}\\sum\\limits_{\\substack{M<m\\le 2M}}\\left| a_m\n\\right|^2,\n\\end{split}\n\\end{equation}\nwhere the star on the sum over $\\chi$ restricts the sum to\nprimitive characters and the asterisks attached to the sum over $m$\nindicates that $m$ runs over square-free integers.\n\\end{theorem}\n\nFollowing the techniques of \\cites{DRHB, DRHB1}, Theorem~\\ref{mainthm} is proved via recursive uses of the Poisson summation formula.   Theorem~\\ref{quarticlargesieve} follows, after some transformations, from Theorem~\\ref{mainthm}.   We note that \\eqref{eq:quartic} is used in \\eqref{square}.   Mark that the characters involved in the second line of \\eqref{square} are actually quadratic, since they are squares of the quartic symbol.  Therefore, it is conceivable that the bounds in \\eqref{final} can be improved if a large sieve inequality for quadratic characters in $\\mathbb{Z}[i]$ is available. \\newline\n\nFinally, we wish to mention that it is highly conceivable that these theorems will find applications in the study of families of $L$-functions involving quartic characters, analogous to those results in \\cite{B&Y} and \\cite{Luo}.\n\n\\subsection{Notations} The following notations and conventions are used throughout the paper.\\\\\n\\noindent $e(z) = \\exp (2 \\pi i z) = e^{2 \\pi i z}$. \\newline\n$\\widetilde{e} (z) = \\exp \\left( 2 \\pi i ( z + \\overline{z} ) \\right)$. \\newline\n$f = O(g)$ or $f \\ll g$ means $|f| \\leq cg$ for some unspecified\npositive constant $c$. \\newline\n\n\\section{Preliminaries}\n\\label{sec 2}\n\\subsection{Quartic symbol and the quartic Gauss sum}\n\\label{sec2.4}\n   The symbol $(\\frac{\\cdot}{n})_4$ is the quartic\nresidue symbol in the ring $\\ensuremath{\\mathbb Z}[i]$.  For a prime $\\pi \\in \\ensuremath{\\mathbb Z}[i]$\nwith $\\mathcal{N}(\\pi) \\neq 2$, the quartic character is defined for $a \\in\n\\ensuremath{\\mathbb Z}[i]$, $(a, \\pi)=1$ by $\\leg{a}{\\pi}_4 \\equiv\na^{(\\mathcal{N}(\\pi)-1)\/4} \\bmod{\\pi}$, with $\\leg{a}{\\pi}_4 \\in \\{\n\\pm 1, \\pm i \\}$. When $\\pi | a$, it is defined that\n$\\leg{a}{\\pi}_4 =0$.  Then the quartic character can be extended\nto composite $n$ with $(\\mathcal{N}(n), 2)=1$ multiplicatively. \\newline\n\n Note that in $\\mathbb{Z}[i]$, the ring of cosets modulo $(1+i)^3$ can be represented by $\\{0, \\pm 1, \\pm i, 1+i, 2, 2i \\}$ and every ideal coprime to $2$ has a unique\ngenerator congruent to 1 modulo $(1+i)^3$ (\\cite[Lemma 7, page\n121]{I&R}). Such a generator is called primary. Recall that the\nquartic reciprocity law states that for two primary primes  $m, n\n\\in \\ensuremath{\\mathbb Z}[i]$,\n\\begin{equation*}\n \\leg{m}{n}_4 = \\leg{n}{m}_4(-1)^{((\\mathcal{N}(n)-1)\/4)((\\mathcal{N}(m)-1)\/4)}.\n\\end{equation*}\n   Observe that a non-unit\n$n=a+bi$ in $\\ensuremath{\\mathbb Z}[i]$ with $a, b \\in \\ensuremath{\\mathbb Z}$ is congruent to 1 modulo $(1+i)^3$ if and only if $a \\equiv 1\n\\bmod{4}, b \\equiv 0 \\bmod{4}$ or $a \\equiv 3 \\bmod{4}, b \\equiv 2\n\\bmod{4}$ by Lemma 6 on page 121 of \\cite{I&R}. \\newline\n\nBelow, we briefly discuss some properties of Gauss sums.  These are dealt with in grater generality in Section 1 of \\cite{Patt}.  For $n , r \\in \\ensuremath{\\mathbb Z}[i]$, $n \\equiv 1 \\bmod {(1+i)^3}$, we set\n\\begin{equation*}\n g(r,n) = \\sum_{x \\bmod{n}} \\leg{x}{n}_4 \\widetilde{e}\\left( \\frac{rx}{n} \\right),\n\\end{equation*}\nwhere here and after\n\\begin{equation} \\label{etildedef}\n\\widetilde{e} (z) = \\exp \\left( 2 \\pi i ( z + \\overline{z} ) \\right).\n\\end{equation}\n   The quartic Gauss sum $g(n)$ is then defined to be $g(n)=g(1,n)$. \\newline\n\n   For $(s,n)=1$, we have\n\\begin{equation*}\n g(rs,n) = \\overline{\\leg{s}{n}_4} g(r,n).\n\\end{equation*}\n    It's easy to see that the above equality in fact holds for any $s$ when\n    $\\leg{\\cdot}{n}_4$ is a primitive character. \\newline\n\n   It's well-known that for square-free $n$'s,\n\\[ |g(n)|=\\sqrt{\\mathcal{N}(n)}. \\]\n\nSuppose $n \\equiv \\pm 1 \\bmod{{(1+i)^3}}$ with no\nrational prime divisor, so $(n, \\bar{n}) = 1$. Let $\\chi_n$ be a\nmultiplicative character on $\\ensuremath{\\mathbb Z}[i]\/(n)$, we define\n\\begin{equation}\n\\label{tau}\n \\tau(\\chi_n) =  \\sum_{1 \\leq x \\leq \\mathcal{N}(n)} \\chi_n(x) e \\left( \\frac{x}{\\mathcal{N}(n)} \\right).\n\\end{equation}\n   Now we specify $\\chi_n$ to be $\\left(\\frac{\\cdot}{n}\\right)_4$. On writing $x = y \\bar{n} + \\bar{y} n$, where $y$ varies over a set\nof representatives in $\\ensuremath{\\mathbb Z}[i] \\bmod{n}$, with $\\bar{n}$ being the\ncomplex conjugate of $n$, it's easy to see that\n\\begin{equation*}\n \\tau(\\chi_n) =  \\sum_{y \\bmod{n}} \\left(\\frac{y \\bar{n}}{n}\\right)_4 e \\left( \\frac{y}{n} + \\frac{\\bar{y}}{\\bar{n}} \\right) =\n \\left(\\frac{\\bar{n}}{n}\\right)_4g(n).\n\\end{equation*}\n   It follows that for $(n_1, n_2)=1$,\n\\begin{align*}\n    \\tau(\\chi_{n_1n_2})=\\left(\\frac{\\mathcal{N}(n_2)}{n_1}\\right)_4\\left(\\frac{\\mathcal{N}(n_1)}{n_2}\\right)_4\\tau(\\chi_{n_1})\\tau(\\chi_{n_2}),\n\\end{align*}\n    and that if $n$ is square-free\n\\begin{align*}\n  |\\tau(\\chi_n)|=\\sqrt{\\mathcal{N}(n)}.\n\\end{align*}\n  Similarly, we have for $n$ square-free\n\\begin{align*}\n  |\\tau(\\chi^2_n)|=\\sqrt{\\mathcal{N}(n)}.\n\\end{align*}\n\n\n\\subsection{Primitive quartic Dirichlet characters}\n\\label{sec2.5}\n   The classification of all the primitive cubic characters of conductor $q$ coprime to $3$ is given in\n   \\cite{B&Y}.\nSimilarly, one can give a classification of all the primitive\nquartic characters of conductor $q$ coprime to\n    $2$.  Every such character is of the form $m \\rightarrow\n(\\frac{m}{n})_4$ for some $n \\in \\ensuremath{\\mathbb Z}[i]$, with $n \\equiv 1\n\\bmod{(1+i)^3}$, $n$ square-free and not divisible by any rational\nprimes and $\\mathcal{N}(n) = q$.\n\n\n\n\\section{Strategy for the proof of Theorem \\ref{mainthm}}\n\n   Our proof of Theorem \\ref{mainthm} uses the ideas in \\cites{DRHB, DRHB1}. We first\n   estimate\n\\begin{equation*}\n   {\\sum}_1=\\sideset{}{^*}\\sum_{\\substack{m \\in \\ensuremath{\\mathbb Z}[i] \\\\M < \\mathcal{N}(m) \\leq 2M}} \\left| \\ \\sideset{}{^*}\\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i] \\\\N < \\mathcal{N}(n) \\leq 2N}} a_n \\leg{n}{m}_4\n   \\right|^2.\n\\end{equation*}\n   We further simplify notation by supposing that\nthe coefficients $a_n$ are supported on such integers $n \\in\n\\ensuremath{\\mathbb Z}[i]$ satisfying $N < \\mathcal{N}(n)  \\leq 2N$.\n   We begin by defining the norm\n\\begin{equation*}\n   \\mathcal{B}_1(M,N)=\\sup \\left\\{ {\\sum}_1: \\sum_n|a_n|^2=1 \\right\\}.\n\\end{equation*}\nTherefore, we need to show\n\\begin{equation*}\n   \\mathcal{B}_1(M,N) \\ll_{\\varepsilon} (MN)^{\\varepsilon} \\left( M + N + (MN)^{3\/4} \\right).\n\\end{equation*}\nIntroducing a smooth weight function, we have\n\\begin{equation*}\n   {\\sum}_1 \\ll \\sum_m \\exp \\left (-2\\pi \\frac{\\mathcal{N}(m)}{M} \\right) \\left| \\ \\sideset{}{^*}\\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i] \\\\ N <\\mathcal{N}(n) \\leq 2N}} a_n \\leg{n}{m}_4\n   \\right|^2,\n\\end{equation*}\n   the sum being over all $m \\in \\ensuremath{\\mathbb Z}[i]$ for which $m \\equiv 1 \\bmod {(1+i)^3}$. If we now expand\nthe above expression we obtain sums of the form\n\\begin{equation} \\label{2.1}\n   \\sum_m \\exp \\left( -2\\pi \\frac{\\mathcal{N}(m)}{M} \\right) \\leg{n_1}{m}_4\\overline{\\leg{n_2}{m}}_4.\n\\end{equation}\n   We note the following analogue of Lemma 2 of Heath-Brown and Patterson\n   \\cite{H&P}.  As the proof is similar, we omit it here.\n\\begin{lemma}\n\\label{lem1} Let $\\chi$ be a character of modulus $f \\neq 1$, not\nnecessarily primitive. Then, for $w \\leq 1, \\varepsilon>0$,\n\\begin{equation*}\n  \\theta(w, \\chi)=\\sum_{\\substack{ a \\equiv 1 \\bmod{ (1+i)^3} \\\\ (a,f)=1}} \\chi(a)e^{-2\\pi \\mathcal{N}(a)w} \\ll E(\\chi)w^{-1}+\\mathcal{N}(f)^{1\/2+\\varepsilon},\n\\end{equation*}\n  where $E(\\chi)=1$ if $\\chi$ is principal, $0$ otherwise. The\n  implied constant depends only on $\\varepsilon$.\n\\end{lemma}\n\nLemma \\ref{lem1} implies that each of these sums in \\eqref{2.1}\nare $O\\left( \\mathcal{N}(n_1n_2)^{1\/2+\\varepsilon} \\right)$, provided that the\ncharacter involved is non-principal. Since $n_1$ and $n_2$ are\nsquare-free, $\\leg{n_1}{m}_4\\overline{\\leg{n_2}{m}_4}$ is principle only if  $n_1 = n_2$.\nIt follows that\n\\[ {\\sum}_1 \\ll_{\\varepsilon}\n   N^{\\varepsilon} \\left( M\\sum_n|a_n|^2+N\\sum_{n_1,n_2}|a_{n_1}a_{n_2}| \\right)  \\ll_{\\varepsilon}\n   N^{\\varepsilon} \\left( M+N^2 \\right)\\sum_n|a_n|^2. \\]\n We therefore have\n\\begin{equation}\n\\label{initialest}\n   \\mathcal{B}_1(M,N) \\ll_{\\varepsilon} N^{\\varepsilon}\\left( M+N^2 \\right).\n\\end{equation}\n    This will be the starting point for an iterative bound for\n    $\\mathcal{B}_1(M,N)$. \\newline\n\n   Similar to the proof of \\cite[Lemma 1]{DRHB}, using the duality\n   principle (see for example, \\cite[Chap. 9]{HM}) and the quartic\n   reciprocity law by considering the case for $n=a+bi$ with\n   $a \\equiv 1 \\bmod{4}, b \\equiv 0 \\bmod{4}$ or $a\n\\equiv 3 \\bmod{4}, b \\equiv 2 \\bmod{4}$ (and similarly for $m$),\nwe can establish the following lemma.\n\\begin{lemma}\n\\label{lem2} We have $\\mathcal{B}_1(M,N) \\leq 2\n\\mathcal{B}_1(N,M)$. Moreover, there exist coefficients $a'_n,\na''_n$ with $|a'_n|=|a''_n|=|a_n|$ such that\n\\begin{equation*}\n\\begin{split}\n\\sideset{}{^*}\\sum_{\\substack{m \\in \\ensuremath{\\mathbb Z}[i] \\\\M <\\mathcal{N}(m) \\leq 2M}} \\left| \\\n\\sideset{}{^*}\\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i] \\\\N < \\mathcal{N}(n) \\leq 2N}} a_n\n\\leg{n}{m}_4\n   \\right|^2  & \\leq 2 \\sideset{}{^*}\\sum_{\\substack{m \\in \\ensuremath{\\mathbb Z}[i] \\\\M< \\mathcal{N}(m) \\leq 2M}} \\left| \\ \\sideset{}{^*}\\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i] \\\\N < \\mathcal{N}(n) \\leq 2N}} a'_n \\leg{m}{n}_4\n   \\right|^2 \\\\\n   & \\leq 4 \\sideset{}{^*}\\sum_{\\substack{m \\in \\ensuremath{\\mathbb Z}[i] \\\\M< \\mathcal{N}(m) \\leq 2M}} \\left| \\ \\sideset{}{^*}\\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i] \\\\N < \\mathcal{N}(n) \\leq 2N}} a''_n \\leg{n}{m}_4\n   \\right|^2.\n\\end{split}\n\\end{equation*}\n\\end{lemma}\n\n\n    Our next lemma is a trivial modification of Lemma 9 of \\cite{DRHB}, which shows that the norm $\\mathcal{B}_1(M,N)$ is essentially\n    increasing.\n\\begin{lemma}\n\\label{lem3} There is an absolute constant $C > 0$ as follows.\nLet $M_1, N \\geq 1$ and $M_2 \\geq CM_1\\log(2M_1N)$. Then\n\\begin{equation*}\n   \\mathcal{B}_1(M_1,N)\\leq C \\mathcal{B}_1(M_2,N).\n\\end{equation*}\n   Similarly, if $M, N_1 \\geq 1$ and $N_2 \\geq CN_1\\log(2N_1M)$. Then\n\\begin{equation*}\n   \\mathcal{B}_1(M,N_1)\\leq C \\mathcal{B}_1(M,N_2).\n\\end{equation*}\n\\end{lemma}\n\nNext, we define\n\\begin{equation*}\n   \\mathcal{B}_2(M,N)=\\sup \\left\\{ {\\sum}_2: \\sum_n|a_n|^2=1 \\right\\},\n\\end{equation*}\n    where\n\\begin{equation}\n\\label{sum2}\n   {\\sum}_2=\\sum_{\\substack{m \\in \\ensuremath{\\mathbb Z}[i] \\\\M < \\mathcal{N}(m) \\leq 2M}} \\left| \\ \\sideset{}{^*}\\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i] \\\\N < \\mathcal{N}(n) \\leq 2N}} a_n \\leg{m}{n}_4\n   \\right|^2,\n\\end{equation}\n   the summation over $m$ running over all integers of $\\ensuremath{\\mathbb Z}[i]$ in the relevant range. \\newline\n\n   It follows directly from Lemma \\ref{lem2} that\n\\begin{equation}\n\\label{12comparison}\n  \\mathcal{B}_1(M,N) \\leq 2\\mathcal{B}_2(M,N).\n\\end{equation}\nFor the other direction, we have the following.\n\\begin{lemma}\n\\label{lem4} There exist $X, Y \\gg 1$ such that $XY^3 \\ll M$ and\n\\begin{equation*}\n   \\mathcal{B}_2(M,N) \\ll (\\log M)^3M^{1\/2}X^{-1\/2}Y^{-3\/2}\\min(Y\\mathcal{B}_1(X,N), X\\mathcal{B}_1(Y,N)).\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\n   To handle $\\sum_2$ we write each of the integers $m$ occurring in the outer summation\nof \\eqref{sum2} in the form $m = ab^2c^3d$, where $a, b, c\\equiv 1\n\\bmod {(1+i)^3}$ are square-free, and $d$ is a product of a unit,\na power of $1+i$, and a fourth power (so that $d$ can be written\nas $d=u(1+i)^je^4$ where $u$ is a unit, $0 \\leq j \\leq 3$ and $e\n\\in \\ensuremath{\\mathbb Z}[i]$). We split the available ranges for $a, b, c$ and $d$\ninto sets $X < \\mathcal{N}(a) \\leq 2X, Y < \\mathcal{N}(b) \\leq 2Y, Z < \\mathcal{N}(c) \\leq 2Z$\nand $W < \\mathcal{N}(d) \\leq 2W$, where $X, Y, Z$ and $W$ are powers of $2$.\nThere will therefore be $O(\\log^3M)$ possible quadruples $X, Y,Z,\nW$. We may now write\n\\begin{equation*}\n   {\\sum}_2 \\ll \\sum_{X,Y,Z, W}{\\sum}_2(X,Y,Z,W)\n\\end{equation*}\n  accordingly, so that\n\\begin{equation*}\n   {\\sum}_2 \\ll (\\log^3M){\\sum}_2(X,Y,Z,W)\n\\end{equation*}\n   for some quadruple $X,Y,Z, W$. However,\n\\begin{equation*}\n   {\\sum}_2(X,Y,Z,W) \\leq \\sum_{b,c,d}\\sideset{}{^*}\\sum_{\\substack{a  \\in \\ensuremath{\\mathbb Z}[i] \\\\X'<\\mathcal{N}(a) \\leq 2X'}}\n   \\left| \\ \\sideset{}{^*}\\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i] \\\\N < \\mathcal{N}(n) \\leq 2N}} a_n \\leg{b^2c^3d}{n}_4 \\leg{a}{n}_4\n   \\right|^2,\n\\end{equation*}\n   where $X'=X'(b,c, d)=M\/\\mathcal{N}(b^2c^3d)$. It is easy to see that $X\/2 \\leq X' \\leq\n   2X$, and hence by Lemma \\ref{lem2}\n\\[ {\\sum}_2(X,Y,Z, W) \\ll \\sum_{b,c,d}\\mathcal{B}_1(X',N)\\sum_{n}\n   |a_n|^2 \\ll YZW^{1\/4}\\max \\left\\{ \\mathcal{B}_1(X',N): X\/2 \\leq X' \\leq 2X \\right\\} \\sum_{n}\n   |a_n|^2, \\]\n since there are $O(W^{1\/4})$ possible integers $d$. \\newline\n\n   In the same way we have\n\\begin{align*}\n   {\\sum}_2(X,Y,Z, W) &\\leq \\sum_{a, b, d}\\sideset{}{^*}\\sum_{\\substack{c  \\in \\ensuremath{\\mathbb Z}[i] \\\\Z'<\\mathcal{N}(c) \\leq 2Z'}}\n   \\left| \\ \\sideset{}{^*}\\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i] \\\\N < \\mathcal{N}(n) \\leq 2N}} a_n \\leg {ab^2d}{n}_4 \\leg {c^3}{n}_4\n   \\right|^2 \\\\\n   &= \\sum_{a,b, d}\\sideset{}{^*}\\sum_{\\substack{c  \\in \\ensuremath{\\mathbb Z}[i] \\\\Z'<\\mathcal{N}(c) \\leq 2Z'}}\n   \\left| \\ \\sideset{}{^*}\\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i] \\\\N < \\mathcal{N}(n) \\leq 2N}} \\overline{a}_n \\overline{\\leg {ab^2d}{n}}_4 \\overline{\\leg {c^3}{n}}_4\n   \\right|^2 \\\\\n   &= \\sum_{a,b,d}\\sideset{}{^*}\\sum_{\\substack{c  \\in \\ensuremath{\\mathbb Z}[i] \\\\Z'<\\mathcal{N}(c) \\leq 2Z'}}\n   \\left| \\ \\sideset{}{^*}\\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i] \\\\N < \\mathcal{N}(n) \\leq 2N}} \\overline{a}_n \\overline{\\leg {ab^2d}{n}}_4 \\leg {c}{n}_4\n   \\right|^2 \\\\\n   & \\ll \\ \\sum_{a,b,d}\\mathcal{B}_1(Z',N)\\sum_{n}\n   |a_n|^2 \\\\\n   & \\ll XYW^{1\/4}\\max \\left\\{ \\mathcal{B}_1(Z',N): Z \\ll Z' \\ll Z \\right\\} \\sum_{n}\n   |a_n|^2,\n\\end{align*}\n   where $Z'=Z'(a,b,d)=M\/\\mathcal{N}(ab^2d)$. As $Y \\ll M^{1\/2}X^{-1\/2}Z^{-3\/2}W^{-1\/2}$, we see that\n\\begin{equation*}\n   \\mathcal{B}_2(M,N) \\ll (\\log M)^3M^{1\/2}X^{-1\/2}Z^{-3\/2}W^{-1\/4}\\min(Z\\mathcal{B}_1(X,N), X\\mathcal{B}_1(Z,N)).\n\\end{equation*}\n   The assertion of the lemma now follows on replacing $Z$ by $Y$ above.\n\\end{proof}\n\n   As in \\cite{DRHB}, we introduce an infinitely differentiable weight function $W: \\ensuremath{\\mathbb R} \\rightarrow \\ensuremath{\\mathbb R}$, defined by\n\\begin{equation}\n\\label{W}\nW(x)= \\begin{cases} \\exp\\left(\\frac{-1}{(2x-1)(5-2x)}\\right),\n\\qquad & \\text{if }  \\frac{1}{2}<x<\\frac{5}{2},\\\\  0, \\qquad  &\n\\text{otherwise}. \\end{cases}\n\\end{equation}\nWe now have\n\\begin{equation}\n\\label{3.7}\n   {\\sum}_2 \\ll \\sum_{m \\in \\ensuremath{\\mathbb Z}[i]}W\\left( \\frac {\\mathcal{N}(m)}{M}\\right) \\left| \\sum_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i] \\\\N < \\mathcal{N}(n) \\leq 2N}} a_n \\leg {m}{n}_4\n   \\right|^2,\n\\end{equation}\n   where we recall that we can drop the conditions on $a_n$ on the inner sum above of the right-hand side expression by supposing that\nthe coefficients $a_n$ are supported on square-free integers $n\n\\equiv 1 \\bmod {(1+i)^3} \\in \\ensuremath{\\mathbb Z}[i]$ lying in the range $N < \\mathcal{N}(n)\n\\leq 2N$. \\newline\n\nExpanding the sum on the right-hand side of \\eqref{3.7}, we obtain\n\\begin{equation*}\n   {\\sum}_2 \\ll \\sum_{n_1,n_2}a_{n_1}\\overline{a}_{n_2}\\sum_{m \\in \\ensuremath{\\mathbb Z}[i]}\n   W\\left(\\frac {\\mathcal{N}(m)}{M}\\right)\n   \\leg{m}{n_1}_4\\overline{\\leg{m}{n_2}}_4.\n\\end{equation*}\n   We set\n\\begin{equation*}\n   {\\sum}_3={\\sum}_3(M,N)= \\sum_{(n_1,n_2)=1}a_{n_1}\\overline{a}_{n_2}\\sum_{m \\in \\ensuremath{\\mathbb Z}[i]}\n   W\\left(\\frac {\\mathcal{N}(m)}{M}\\right)\n   \\leg{m}{n_1}_4\\overline{\\leg{m}{n_2}}_4\n\\end{equation*}\n   and define\n\\begin{equation} \\label{B3def}\n   \\mathcal{B}_3(M,N)=\\sup \\left\\{ {\\sum}_3: \\sum_n|a_n|^2=1 \\right\\}.\n\\end{equation}\n    Similar to \\cite[Lemma 7]{DRHB, DRHB1}, we have the\n    following\n\\begin{lemma}\n\\label{lem5} Let $\\varepsilon > 0$ be given. Then there exist\npositive integers $\\Delta_2 \\geq \\Delta_1$ such that\n\\begin{equation*}\n   \\mathcal{B}_2(M,N) \\ll_{\\varepsilon} N^{\\varepsilon} \\mathcal{B}_3 \\left(\\frac {M}{\\Delta_1}, \\frac {N}{\\Delta_2}\\right).\n\\end{equation*}\n\\end{lemma}\n\n\n   We complete the chain of relations amongst the various norms by giving the following estimate for $\\mathcal{B}_3(M,N)$ in terms of\n   $\\mathcal{B}_2(M,N)$.\n\\begin{lemma}\n\\label{lem6} Let $N \\geq 1$. Then for any $\\varepsilon>0$ we have\n\\begin{equation*}\n   \\mathcal{B}_3(M,N) \\ll_{\\varepsilon} MN^{4\\varepsilon-1}\\max \\left\\{ \\mathcal{B}_2(K,\n   N): K \\leq N^2\/M\n   \\right\\}+M^{-1}N^{3+4\\varepsilon}\\sum_{K>N^2\/M}K^{-2-\\varepsilon}\\mathcal{B}_2(K,\n   N),\n\\end{equation*}\n   where $K$ runs over powers of $2$.\n\\end{lemma}\nThis bound uses the Poisson summation formula and is the key\nin the proof of Theorem \\ref{mainthm}.  Note that it does not cover the case in which $N =1\/2$, say, for which we have the trivial bound\n\\begin{equation}\n\\label{trivialbound}\n   \\mathcal{B}_3(M,N) \\ll_{\\varepsilon} M, \\hspace{0.1in} (N \\leq 1).\n\\end{equation}\nSection \\ref{sec4} will be devoted to the proof of Lemma~\\ref{lem6}.\n\n\\section{Proof of Lemma \\ref{lem6}} \\label{sec4}\n\nOur proof of Lemma \\ref{lem6} requires the application of the Poisson\nsummation formula. We shall write\n\\begin{equation*}\n   \\chi(m)=\\leg{m}{n_1}_4\\overline{\\leg{m}{n_2}}_4,\n\\end{equation*}\nwhich is a primitive character (on the group\n$(\\ensuremath{\\mathbb Z}[i]\/(n_1n_2))^{\\times}$) to modulus $q = n_1n_2$, provided\nthat $n_1$, $n_2$ and 2 are pair-wise coprime and that $n_1$ and $n_2$ are\nsquare-free.\n\\begin{lemma}\n\\label{lem7} With the above notations we have\n\\begin{equation*}\n   \\sum_{m \\in \\ensuremath{\\mathbb Z}[i]}W\\left(\\frac {\\mathcal{N}(m)}{M}\\right)\\chi(m)=\\frac {\\chi(-2i)g(n_1)\\overline{g(n_2)}M}{\\mathcal{N}(q)}\\leg{n_2}{n_1}_4\\overline{\\leg{n_1}{n_2}}_4\\leg{-1}{n_2}_4\n   \\sum_{m \\in \\ensuremath{\\mathbb Z}[i]}\\widetilde{W}\\left(\\sqrt{\\frac {\\mathcal{N}(k)M}{\\mathcal{N}(q)}}\\right)\\overline{\\chi}(k),\n\\end{equation*}\n   where\n\\begin{equation*}\n   \\widetilde{W}(t)=\\int\\limits^{\\infty}_{-\\infty}\\int\\limits^{\\infty}_{-\\infty}W(\\mathcal{N}(x+yi))\\widetilde{e}\\left( \\frac{t(x+yi)}{2i}\\right)\\mathrm{d} x  \\mathrm{d} y,\n\\end{equation*}\n   for non-negative $t$. Here $\\widetilde{e}(z)$ is defined in \\eqref{etildedef} and $g(n)$ is the Gauss sums defined in Section \\ref{sec2.4}.\n\\end{lemma}\n\\begin{proof}\nThis lemma is analogous to Lemma 10 in \\cite{DRHB1} and the proof is very similar.  The differences include we need to start with the Poisson summation formula for\n   $\\ensuremath{\\mathbb Z}[i]$, which takes the form.\n\\begin{equation*}\n   \\sum_{j \\in \\ensuremath{\\mathbb Z}[i]}f(j)=\\sum_{k \\in\n   \\ensuremath{\\mathbb Z}[i]}\\int\\limits^{\\infty}_{-\\infty}\\int\\limits^{\\infty}_{-\\infty}f(x+yi)\\widetilde{e}\\left( \\frac {k(x+yi)}{2i} \\right)\\mathrm{d} x  \\mathrm{d} y.\n\\end{equation*}\nWe omit the details of the rest of proof as it simply goes along the same lines as the proof of Lemma 10 in \\cite{DRHB1}.\n\\end{proof}\n\n    Our next result will be used to separate the variables in a function of a\nproduct, which is Lemma 12 of \\cite{DRHB}.\n\\begin{lemma}\n\\label{lem8} Let $\\rho: \\ensuremath{\\mathbb R} \\rightarrow \\ensuremath{\\mathbb R}$ be an infinitely\ndifferentiable function whose derivatives satisfy the bound\n\\[ \\rho{(k)}(x) \\ll_{k,A}|x|^{-A} \\]\nfor $|x| \\geq 1$, for any positive constant\n$A$. Let\n\\begin{equation*}\n   \\rho_{+}(s)=\\int\\limits^{\\infty}_{0}\\rho(x)x^{s-1} \\mathrm{d} x,\n   \\hspace{0.1in} \\rho_{-}(s)=\\int\\limits^{\\infty}_{0}\\rho(-x)x^{s-1} \\mathrm{d}\n   x.\n\\end{equation*}\n   Then $\\rho_{+}(s)$ and $\\rho_{-}(s)$ are holomorphic in $\\Re(s) = \\sigma > 0$, and\n   satisfy\n\\begin{equation*}\n   \\rho_{+}(s), \\ \\rho_{-}(s) \\ll_{A, \\sigma} |s|^{-A},\n\\end{equation*}\nin that same domain, for any positive constant $A$. Moreover if $\\sigma> 0$ we have\n\\begin{equation*}\n   \\rho(x)=\\frac {1}{2\\pi\n   i}\\int\\limits^{\\sigma+i\\infty}_{\\sigma-i\\infty}\\rho_{+}(s)x^{-s} \\mathrm{d} s \\; \\; \\; \\mbox{and} \\; \\; \\;\n   \\rho(-x)=\\frac {1}{2\\pi\n   i}\\int\\limits^{\\sigma+i\\infty}_{\\sigma-i\\infty}\\rho_{-}(s)x^{-s} \\mathrm{d} s\n\\end{equation*}\n   for any positive $x$.\n\\end{lemma}\n\nWe are now ready to prove Lemma~\\ref{lem6}.\n\n\\begin{proof} [Proof of Lemma~\\ref{lem6}]\n   In the notation of Lemma \\ref{lem7} we have\n\\begin{equation*}\n   {\\sum}_3(M,N)=\\sum_{(n_1,n_2)=1}a_{n_1}\\overline{a}_{n_2}\\sum_{m \\in \\ensuremath{\\mathbb Z}[i]}W\\left(\\frac {\\mathcal{N}(m)}{M}\\right)\\chi(m).\n\\end{equation*}\n   We proceed to evaluate the inner sum using Lemma \\ref{lem7}, whence\n\\begin{equation}\n\\label{5.1}\n   {\\sum}_3(M,N)=M\\sum_{k \\in \\ensuremath{\\mathbb Z}[i]}\\sum_{(n_1,n_2)=1}c_{n_1}\\overline{c}_{n_2}\\leg{n_2}{n_1}_4\\overline{\\leg{n_1}{n_2}}_4\\leg{-1}{n_2}_4\\widetilde{W}\\left(\\sqrt{\\frac {\\mathcal{N}(k)M}{\\mathcal{N}(n_1n_2)}}\\right)\\overline{\\chi}(k),\n\\end{equation}\n   where\n\\begin{equation*}\n    c_n=a_n\\leg{-2i}{n}_4\\frac {g(n)}{\\mathcal{N}(n)}.\n\\end{equation*}\n   Note by the law of quartic reciprocity, we have\n\\begin{equation*}\n   \\leg{n_2}{n_1}_4\\overline{\\leg{n_1}{n_2}}_4=(-1)^{((\\mathcal{N}(n_1)-1)\/4)((\\mathcal{N}(n_2)-1)\/4)}.\n\\end{equation*}\n    Now we let\n\\[ S_1 =\\{n \\in \\ensuremath{\\mathbb Z}[i]: N<\\mathcal{N}(n)\\leq 2N, n \\hspace{0.05in} \\text{square-free}, n=a+bi, a, b \\in \\ensuremath{\\mathbb Z}, a \\equiv 1 \\bmod{4}, b \\equiv 0 \\bmod{4} \\},  \\]\nand\n\\[ S_2 =\\{n \\in \\ensuremath{\\mathbb Z}[i]: N<\\mathcal{N}(n)\\leq 2N, n \\hspace{0.05in} \\text{square-free}, n=a+bi, a, b \\in \\ensuremath{\\mathbb Z}, a \\equiv 3 \\bmod{4}, b \\equiv 2 \\bmod{4} \\}. \\]\n   We can then recast the inner sum in \\eqref{5.1} as\n\\begin{align*}\n   &\\sum_{(n_1,n_2)=1} \\cdots \\\\\n   =& \\sum_{\\substack{(n_1,n_2)=1 \\\\ n_1 \\in S_1, n_2 \\in S_1}} \\cdots\n   \\; \\; +\\sum_{\\substack{(n_1,n_2)=1 \\\\ n_1 \\in S_1, n_2 \\in S_2}} \\cdots \\; \\; +\\sum_{\\substack{(n_1,n_2)=1 \\\\ n_1 \\in S_2, n_2 \\in S_1}} \\cdots \\; \\;\n   +\\sum_{\\substack{(n_1,n_2)=1 \\\\ n_1 \\in S_2, n_2 \\in S_2}} \\cdots \\; \\; -2\\sum_{\\substack{(n_1,n_2)=1 \\\\ n_1 \\in S_2, n_2 \\in S_2}} \\cdots \\\\\n   =& \\sum_{(n_1,n_2)=1}c_{n_1}\\overline{c}_{n_2}\\leg{-1}{n_2}_4\\widetilde{W}\\left(\\sqrt{\\frac {\\mathcal{N}(k)M}{\\mathcal{N}(n_1n_2)}}\\right)\\overline{\\chi}(k)-2\\sum_{(n_1,n_2)=1}c'_{n_1}\\overline{c'}_{n_2}\\leg{-1}{n_2}_4\\widetilde{W}\\left(\\sqrt{\\frac {\\mathcal{N}(k)M}{\\mathcal{N}(n_1n_2)}}\\right)\\overline{\\chi}(k),\n\\end{align*}\n   where we let $c'_n=c_n$ if $n \\in S_2$ and $0$ otherwise. Due\n   to similarities, it suffices to estimate\n\\begin{equation*}\n   M\\sum_{k \\in\n\\ensuremath{\\mathbb Z}[i]}\\sum_{(n_1,n_2)=1}c_{n_1}\\overline{c}_{n_2}\\leg{-1}{n_2}_4\\widetilde{W}\\left(\\sqrt{\\frac\n{\\mathcal{N}(k)M}{\\mathcal{N}(n_1n_2)}}\\right)\\overline{\\chi}(k),\n\\end{equation*}\n   Note that $k = 0$ may be omitted if $N \\geq 1$, since then $\\mathcal{N}(n_1n_2) > 1$ and $\\chi(0) = 0$,\nthe character being non-trivial. We may now apply Lemma \\ref{lem8}\nto the function $\\rho(x) = \\widetilde{W}(x)$, which satisfies the necessary conditions of the lemma, as one sees by repeated\nintegration by parts. We decompose the available $k$ into sets for\nwhich $K < \\mathcal{N}(k) \\leq 2K$, where $K$ runs over powers of $2$, and\nuse\n\\begin{equation} \\label{sigmadef}\n\\sigma = \\left\\{ \\begin{array}{ll} \\varepsilon , &  \\mbox{for} \\;  K \\leq N^2\/M, \\\\  4 +\n\\varepsilon , &  \\mbox{otherwise}.\n\\end{array} \\right.\n\\end{equation}\nThis gives\n\\begin{align*}\n    {\\sum}_3 & \\ll_{\\varepsilon} M\n    \\sum_{K}(KM)^{-\\sigma\/2}\\int\\limits^{\\infty}_{-\\infty}|\\rho_{+}(\\sigma+it)||S(\\sigma+it)| \\mathrm{d} t,\n\\end{align*}\n  where\n\\[\n   S(s)=\\sum_{K < \\mathcal{N}(k) \\leq 2K}\\left| \\sum_{(n_1,n_2)=1}b_{n_1}b'_{n_2}\\leg{-1}{n_2}_4\\overline{\\chi}(k)\\right|, \\; \\; \\;  \\mbox{with} \\; \\; \\;\n    b_n=c_n\\mathcal{N}(n)^{s\/2} \\; \\; \\mbox{and} \\; \\; b'_n=\\overline{c}_n\\mathcal{N}(n)^{s\/2} .\n\\]\nWe use the M\\\"obius function to detect the coprimality condition in the inner sum of $S(s)$, giving\n\\begin{align*}\n S(s)  &\\ll \\sum_{d}\\sum_{K < \\mathcal{N}(k) \\leq 2K}\\left| \\sum_{d|(n_1,n_2)}b_{n_1}b'_{n_2}\\leg{-1}{n_2}_4\\overline{\\chi}(k)\\right| \\\\\n&= \\sum_{d}\\sum_{K < \\mathcal{N}(k) \\leq 2K}\\left|\n\\sum_{d|n}b_{n}\\overline{\\leg{k}{n}}_4\\right| \\left|\n\\sum_{d|n}b'_{n}\\leg{-k}{n}_4\\right| \\leq S^{1\/2}_1S^{1\/2}_2,\n\\end{align*}\n   by Cauchy's inequality, where\n\\[  S_1 = \\sum_{d}\\sum_{K < \\mathcal{N}(k) \\leq 2K}\\left| \\sum_{d|n}b_{n}\\overline{\\leg{k}{n}}_4\\right|^2 \\]\nand satisfies the bound\n\\[  S_1 \\leq \\sum_{d}\\mathcal{B}_2(K,N)\\sum_{d|n}|b_n|^2  \\leq \\mathcal{B}_2(K,N)\\sum_{n}d(n)|a_n|^2\\mathcal{N}(n)^{\\sigma-1} \\ll_{\\varepsilon} N^{\\varepsilon+\\sigma-1}\\mathcal{B}_2(K,N). \\]\n$S_2$ can be treated similarly. It follows then that\n\\begin{equation*}\n   S(s) \\ll_{\\varepsilon} N^{\\varepsilon+\\sigma-1}\\mathcal{B}_2(K,N),\n\\end{equation*}\n   and since\n\\begin{align*}\n     \\int\\limits^{\\infty}_{-\\infty}|\\rho_{+}(\\sigma+it)| \\mathrm{d} t \\ll_{\\varepsilon} 1,\n\\end{align*}\nwe infer, mindful of our choices of $\\sigma$ in \\eqref{sigmadef}, that\n\\begin{align*}\n    {\\sum}_3 & \\ll_{\\varepsilon}  MN^{4\\varepsilon-1}\\max \\left\\{ \\mathcal{B}_2(K,\n   N): K \\leq N^2\/M\n   \\right\\}+M^{-1}N^{3+4\\varepsilon}\\sum_{K>N^2\/M}K^{-2-\\varepsilon}\\mathcal{B}_2(K,\n   N),\n\\end{align*}\nRecalling the definition of $\\mathcal{B}_3(M,N)$ in \\eqref{B3def}, we have completed the proof of Lemma \\ref{lem6}.\n   \\end{proof}\n\n\\section{The Recursive Estimate and the Proof of Theorem~\\ref{mainthm}}\n\n    Lemmas \\ref{lem4}, \\ref{lem5} and \\ref{lem6} allow us to estimate $\\mathcal{B}_1(M,N)$ recursively, as follows.\n\\begin{lemma}\n\\label{lem9} Suppose that $3\/2 < \\xi \\leq 2$, and that\n\\begin{equation}\n\\label{6.1}\n   \\mathcal{B}_1(M,N)\\ll_{\\varepsilon} (MN)^{\\varepsilon} \\left( M+N^{\\xi} + (MN)^{3\/4} \\right).\n\\end{equation}\n   for any $\\varepsilon > 0$. Then\n\\begin{equation*}\n   \\mathcal{B}_1(M,N)\\ll_{\\varepsilon} (MN)^{\\varepsilon} \\left( M+N^{(9\\xi-6)\/(4\\xi-1)} + (MN)^{3\/4} \\right).\n\\end{equation*}\n   for any $\\varepsilon > 0$.\n\\end{lemma}\n\\begin{proof}\n  By the symmetry expressed in Lemma \\ref{lem2} the hypothesis\n\\eqref{6.1} yields\n\\begin{align*}\n     \\mathcal{B}_1(M,N)\\ll_{\\varepsilon} \\left( M^{\\xi} + N + (MN)^{3\/4}\\right) (MN)^{\\varepsilon}.\n\\end{align*}\n    It follows from \\eqref{initialest} that the above estimation is valid with\n    $\\xi=2$. We now feed this into Lemma \\ref{lem4}, whence\n\\begin{equation} \\label{eqwithmin}\n   \\mathcal{B}_2(M,N) \\ll_{\\varepsilon} (MN)^{2\\varepsilon}M^{1\/2}X^{-1\/2}Y^{-3\/2}\\min(Yf(X,N),\n   Xf(Y,N)),\n\\end{equation}\n   where\n\\begin{equation*}\n     f(Z,N)=Z^{\\xi} + N + (ZN)^{3\/4}.\n\\end{equation*}\n   If $X \\geq Y$ we bound the minimum in \\eqref{eqwithmin} by $Yf(X,N)$, whence\n\\begin{equation*}\n   \\mathcal{B}_2(M,N) \\ll_{\\varepsilon} (MN)^{2\\varepsilon}M^{1\/2}X^{-1\/2}Y^{-3\/2}\\left( YX^{\\xi} + YN +\n   Y(XN)^{3\/4} \\right).\n\\end{equation*}\n   Here we have\n\\begin{equation*}\n   M^{1\/2}X^{-1\/2}Y^{-3\/2}YX^{\\xi} \\ll M^{\\xi}Y^{1-3\\xi}\n\\end{equation*}\n   since $X \\ll MY^{-3}$. On recalling that $\\xi > 3\/2 >1\/3$ and $Y \\gg 1$ we see that\nthis is $O(M^{\\xi})$. Moreover\n\\begin{equation*}\n   M^{1\/2}X^{-1\/2}Y^{-3\/2}YN=M^{1\/2}X^{-1\/2}Y^{-1\/2}N \\ll\n   M^{1\/2}N.\n\\end{equation*}\n   Finally\n\\[  M^{1\/2}X^{-1\/2}Y^{-3\/2}Y(XN)^{3\/4}=M^{1\/2}X^{1\/4}Y^{-1\/2}N^{3\/4}\n   \\ll M^{3\/4}N^{3\/4}\n   \\ll M^{1\/2}N+M^{3\/2} \\leq\n   M^{1\/2}N+M^{\\xi}, \\]\n   since $\\xi > 3\/2$. It follows that\n\\begin{equation}\n\\label{B2bound}\n   \\mathcal{B}_2(M,N) \\ll_{\\varepsilon}\n   (MN)^{2\\varepsilon} \\left( M^{1\/2}N+M^{\\xi} \\right)\n\\end{equation}\n   when $X \\geq Y$. \\newline\n\n   In the alternative case we bound the minimum in \\eqref{eqwithmin} by $Xf(Y,N)$, whence\n\\begin{equation*}\n   \\mathcal{B}_2(M,N) \\ll_{\\varepsilon} (MN)^{2\\varepsilon}M^{1\/2}X^{-1\/2}Y^{-3\/2} \\left( XY^{\\xi} + XN + X(YN)^{3\/4} \\right).\n\\end{equation*}\n   Here\n\\begin{equation*}\n   M^{1\/2}X^{-1\/2}Y^{-3\/2}XY^{\\xi} \\ll M^{1\/2}X^{1\/2}Y^{1\/2} \\ll M\n   \\ll M^{\\xi}\n\\end{equation*}\n   since $\\xi \\leq 2$ and $XY \\ll M$. Moreover\n\\begin{equation*}\n   M^{1\/2}X^{-1\/2}Y^{-3\/2}XN  = M^{1\/2}X^{1\/2}Y^{-3\/2}N \\ll\n   M^{1\/2}N\n\\end{equation*}\n  since we are now supposing that $Y \\geq X$.\n  Finally\n\\[ M^{1\/2}X^{-1\/2}Y^{-3\/2}X(YN)^{3\/4}=M^{1\/2}X^{1\/2}Y^{-3\/4}N^{3\/4} \\ll\n   M^{1\/2}Y^{-1\/4}N^{3\/4} \\ll  M^{1\/2}N^{3\/4} \\ll M^{1\/2}N+M^{\\xi}, \\]\n   as before. It follows that \\eqref{B2bound} holds when $Y \\geq X$ too. It will be convenient to\nobserve that \\eqref{B2bound} still holds when $M < 1\/2$ , since\nthen $\\mathcal{B}_2(M,N)= 0$. \\newline\n\n   We are now ready to use \\eqref{B2bound} (with a new value for $\\varepsilon$) in Lemma \\ref{lem6}, to obtain\na bound for $\\mathcal{B}_3(M,N)$. We readily see that\n\\begin{equation*}\n   \\max\\left\\{ \\mathcal{B}_2(K,\n   N): K \\leq N^2\/M\n   \\right\\} \\ll_{\\varepsilon} N^{\\varepsilon}\\left( M^{-1\/2}N^2+M^{-\\xi}N^{2\\xi} \\right)\n\\end{equation*}\n   and\n\\begin{equation*}\n   \\sum_{K>N^2\/M}K^{-2-\\varepsilon}\\mathcal{B}_2(K,\n   N) \\ll_{\\varepsilon}\n   N^{\\varepsilon}\\left( M^{3\/2}N^{-2}+M^{2-\\xi}N^{2\\xi-4} \\right).\n\\end{equation*}\n   Thus, if $N \\geq 1$, we will have\n\\begin{equation*}\n    \\mathcal{B}_3(M,\n   N) \\ll_{\\varepsilon}\n   N^{5\\varepsilon}\\left( M^{1\/2}N+M^{1-\\xi}N^{2\\xi-1} \\right).\n\\end{equation*}\n   When this is used in Lemma \\ref{lem5} we find that when $N\/\\Delta_2 \\geq\n   1$,\n\\begin{align*}\n    \\mathcal{B}_3\\left( \\frac {M}{\\Delta_1}, \\frac {N}{\\Delta_2} \\right) &\\ll_{\\varepsilon}\n   N^{5\\varepsilon}\\left( M^{1\/2}N+M^{1-\\xi}N^{2\\xi-1}\\Delta^{\\xi-1}_1\\Delta^{1-2\\xi}_2 \\right) \\leq N^{5\\varepsilon} \\left( M^{1\/2}N+M^{1-\\xi}N^{2\\xi-1}\\Delta^{-\\xi}_2 \\right) \\\\\n   & \\leq N^{5\\varepsilon}\\left( M^{1\/2}N+M^{1-\\xi}N^{2\\xi-1} \\right).\n\\end{align*}\n   Note that when $M \\geq N$, we have $M^{1\/2}N \\leq (MN)^{3\/4}$\n   and when $M \\leq N$, we have $(MN)^{3\/4} \\leq\n   M^{1-\\xi}N^{2\\xi-1}$ since $\\xi > 3\/2$. Thus we conclude that\n\\begin{equation*}\n    \\mathcal{B}_2(M,\n   N) \\ll_{\\varepsilon}\n   N^{6\\varepsilon}\\left( (MN)^{3\/4}+M^{1-\\xi}N^{2\\xi-1} \\right),\n\\end{equation*}\n   provided that $N\/\\Delta_2 \\geq 1$. In the alternative case \\eqref{trivialbound} applies, whence\n\\begin{equation*}\n    \\mathcal{B}_2(M,\n   N) \\ll_{\\varepsilon}\n   (MN)^{6\\varepsilon}\\left( M+(MN)^{3\/4}+M^{1-\\xi}N^{2\\xi-1} \\right),\n\\end{equation*}\n   In view of Lemma \\ref{lem3} and \\eqref{12comparison} we may now deduce that\n\\begin{align*}\n    \\mathcal{B}_1(M,\n   N)  \\leq \\mathcal{B}_1(M',\n   N) \\ll \\mathcal{B}_2(M',\n   N) \\ll_{\\varepsilon}\n   (M'N)^{6\\varepsilon}\\left( M'+(M'N)^{3\/4}+{M'}^{1-\\xi}N^{2\\xi-1} \\right) ,\n\\end{align*}\n  for any $M' \\geq CM \\log(2MN)$. Note that when ${M}^{4\\xi-1}\n  \\leq N^{8\\xi-7}$, we have\n\\begin{align*}\n    (MN)^{3\/4} \\leq {M}^{1-\\xi}N^{2\\xi-1}.\n\\end{align*}\n   We shall now choose\n\\begin{align*}\n    M'=C\\max \\left\\{ M, N^{(8\\xi-7)\/(4\\xi-1)} \\right\\} \\log(2MN) ,\n\\end{align*}\n   so that when $M \\geq N^{(8\\xi-7)\/(4\\xi-1)}$, we have\n\\begin{align*}\n     M'+(M'N)^{3\/4}+{M'}^{1-\\xi}N^{2\\xi-1} \\ll (MN)^{\\varepsilon} \\left( M+(MN)^{3\/4} \\right) ,\n\\end{align*}\n  while when $M \\leq N^{(8\\xi-7)\/(4\\xi-1)}$, we have\n\\begin{align*}\n    M'+(M'N)^{3\/4}+{M'}^{1-\\xi}N^{2\\xi-1} &\n     \\ll (MN)^{\\varepsilon} \\left( N^{(8\\xi-7)\/(4\\xi-1)}+N^{(8\\xi-7)(1-\\xi)\/(4\\xi-1)}N^{2\\xi-1} \\right) \\\\\n     & \\ll\n     (MN)^{\\varepsilon}N^{(9\\xi-6)\/(4\\xi-1)}.\n\\end{align*}\n   We then deduce that\n\\begin{align*}\n    \\mathcal{B}_1(M,\n   N)   \\ll_{\\varepsilon}\n   (MN)^{20\\varepsilon} \\left( M+(MN)^{3\/4}+N^{(9\\xi-6)\/(4\\xi-1)} \\right).\n\\end{align*}\n  Lemma \\ref{lem9} now follows.\n\\end{proof}\n\nWe now proceed to prove Theorem \\ref{mainthm}.\n\n\\begin{proof}[Proof of Theorem~\\ref{mainthm}] Note that it follows from \\eqref{initialest} that the estimation given in Lemma \\ref{lem9} is valid with\n    $\\xi=2$.  We further observe that\n\\begin{align*}\n   \\frac{3}{2} <  \\frac {9\\xi-6}{4\\xi-1} < \\xi,\n\\end{align*}\n  for $\\xi > 3\/2$ and in the iterative applications of Lemma~\\ref{lem9} the exponent of $N$ in the bound for $\\mathcal{B}_1(M,N)$ decreases and tends to $3\/2$.  We therefore\narrive at the following bound\n\\begin{align*}\n    \\mathcal{B}_1(M,\n   N)   \\ll_{\\varepsilon}\n   (MN)^{\\varepsilon}\\left( M+N^{3\/2}+(MN)^{3\/4} \\right),\n\\end{align*}\n   for any $\\varepsilon>0$.  Using Lemma \\ref{lem2} we then have\n\\begin{equation*}\n\\begin{split}\n    \\mathcal{B}_1(M,\n   N)  &  \\ll_{\\varepsilon}\n   (MN)^{\\varepsilon}\\min \\left\\{ M+N^{3\/2}+(MN)^{3\/4},\n   N+M^{3\/2}+(MN)^{3\/4} \\right\\} \\\\\n   & \\ll_{\\varepsilon} (MN)^{\\varepsilon} \\left(M+N+(MN)^{3\/4} \\right),\n   \\end{split}\n   \\end{equation*}\n   where the last estimation follows since when $N \\leq M, N^{3\/2}=N^{3\/4}N^{3\/4} \\leq (MN)^{3\/4}$ and similarly when $N \\geq M, M^{3\/2} \\leq (MN)^{3\/4}$.\n   This establishes Theorem \\ref{mainthm}.\n   \\end{proof}\n\n\n\\section{The Quartic large sieve for Dirichlet Characters}\n\\label{sec8}\n\nWe now proceed to prove Theorem~\\ref{quarticlargesieve}.  It is easy to reduce the expression on the left-hand side of\n\\eqref{final} to a sum of similar expressions with the additional\nsummation conditions $(q,2)=1$ and $(m,2)=1$ included. Thus it\nsuffices to estimate\n\\begin{equation} \\label{trans}\n\\begin{split}\n\\sum\\limits_{\\substack{Q<q\\le 2Q\\\\ (q,2)=1}}\\\n\\sideset{}{^\\star}\\sum\\limits_{\\substack{\\chi \\bmod q\\\\ \\chi^4=\\chi_0,\n\\chi^2 \\neq \\chi_0}} \\left| \\ \\sideset{}{^*}\\sum\\limits_{\\substack{M<m\\le 2M\\\\\n(m,2)=1}} a_m \\chi(m)\\right|^2 &=\n\\sideset{}{'}\\sum\\limits_{\\substack{n\\in \\ensuremath{\\mathbb Z}[i]\\\\ Q<\\mathcal{N}(n)\\le 2Q\\\\\nn\\equiv 1 \\bmod {(1+i)^3}}} \\left| \\\n\\sideset{}{^*}\\sum\\limits_{\\substack{M< m\\le 2M\\\\ (m,2)=1}} a_m \\chi_n(m)\\right|^2\\\\\n&= \\frac{1}{2}\\sideset{}{'}\\sum\\limits_{\\substack{n\\in \\ensuremath{\\mathbb Z}[i]\\\\\nQ<\\mathcal{N}(n)\\le 2Q\\\\ n\\equiv \\pm 1 \\bmod {(1+i)^3}}} \\left| \\\n\\sideset{}{^*}\\sum\\limits_{\\substack{M<m\\le 2M\\\\ (m,2)=1}} a_m\\chi_n(m)\n\\right|^2,\n\\end{split}\n\\end{equation}\nwhere the apostrophe indicates that $n$ is square-free and has no\nrational prime divisor and $\\chi_n(m)=\\left(\\frac{m}{n}\\right)_4$\nis the quartic residue symbol. We shall use this notation for all\n$n\\in \\ensuremath{\\mathbb Z}[i]$ and $m\\in \\ensuremath{\\mathbb Z}$.\n\n\\subsection{Definition of certain norms}\n  In the following, we shall estimate the expression in the last\nline of \\eqref{trans}. We begin by defining a norm corresponding\nto the double sum in the last line of \\eqref{trans} by\n\\begin{equation*}\nB_1(Q,M):=\\sup\\limits_{(a_m)} \\| a_m \\|^{-2}\n\\sideset{}{'}\\sum\\limits_{\\substack{n \\in \\ensuremath{\\mathbb Z}[i]\\\\ Q<\\mathcal{N}(n)\\le 2Q\\\\\nn\\equiv \\pm 1 \\bmod {(1+i)^3}}} \\left| \\\n\\sideset{}{^*}\\sum\\limits_{\\substack{M<m\\le 2M\\\\ (m,2)=1}} a_m\n\\chi_m(n)\\right|^2,\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\|a_m \\|^2 = \\sum_{m} |a_m|^2,\n\\end{equation*}\nand where by convention we suppose that $(a_m)$ is not identically\nzero. \\newline\n\nWe further define a norm $B_2(Q,M)$ in the same way as $B_1(Q,M)$\nexcept removing the condition that $n$ has no rational prime\ndivisor. Similarly, we define a norm $B_3(Q,M)$ by further\nremoving the condition that $n$ is square-free. \\newline\n\n  We now use the function $W(x)$ defined in \\eqref{W} to see that\n\\begin{equation*}\n  B_3(Q,M) \\ll \\sup\\limits_{(a_m)} \\| a_m \\|^{-2} \\sum\\limits_{n\\in \\ensuremath{\\mathbb Z}[i]}\nW\\left(\\frac{\\mathcal{N}(n)}{Q}\\right) \\left| \\\n\\sideset{}{^*}\\sum\\limits_{\\substack{M<m\\le 2M\\\\ (m,2)=1}} a_m\n\\chi_m(n)\\right|^2.\n\\end{equation*}\nExpanding the sum on the right-hand side, we obtain\n\\begin{equation} \\label{db3bound}\n\\sideset{}{^*}\\sum\\limits_{\\substack{M<m_1\\le 2M\\\\ M<m_2\\le 2M\\\\\n(m_1m_2,2)=1}} a_{m_1}\\overline{a}_{m_2} \\sum\\limits_{n\\in \\ensuremath{\\mathbb Z}[i]}\nW\\left(\\frac{\\mathcal{N}(n)}{Q}\\right)\n\\chi_{m_1}(n)\\overline{\\chi}_{m_2}(n).\n\\end{equation}\n\nAs in  \\cite{DRHB}, it turns out that it suffices to restrict our attention to\nthe case in which $m_1$ and $m_2$ are coprime.  To see this, we sort the terms\nin \\eqref{db3bound} according to $\\delta = (m_1,m_2)$, and\ndetecting the condition $(n,\\delta)=1$ by the M\\\"obius function\nfor $\\ensuremath{\\mathbb Z}[i]$, we obtain that the expression in \\eqref{db3bound}\nequals\n\\begin{equation*}\n\\begin{split}\n\\sideset{}{^*}\\sum_{(\\delta,2)=1} \\sum\\limits_{d|\\delta} \\mu_{[i]}(d)\n\\sum\\limits_{s\\in \\ensuremath{\\mathbb Z}[i]}\nW\\left(\\frac{\\mathcal{N}(s)}{Q\/\\mathcal{N}(d)}\\right) & \\sideset{}{^*}\\sum\\limits_{\\substack{M\/\\delta<r_1\\le\n2M\/\\delta \\\\ M\/\\delta<r_2\\le 2M\/\\delta\\\\ (r_1r_2,2\\delta)=1\\\\\n(r_1,r_2)=1}} a_{r_1}^*\\overline{a_{r_2}^*}\n\\chi_{r_1}(s)\\overline{\\chi_{r_2}}(s) \\\\\n& \\ll  \\sideset{}{^*}\\sum_{(\\delta,2)=1} \\sum\\limits_{d|\\delta}\nB_4\\left(\\frac{Q}{\\mathcal{N}(d)},\\frac{M}{\\delta}\\right)\n\\sideset{}{^*}\\sum\\limits_{\\substack{M\/\\delta<r\\le 2M\/\\delta \\\\\n(r,2\\delta)=1}} |a_{r\\delta}|^2,\n\\end{split}\n\\end{equation*}\nwhere $d$ runs over non-associate divisors of $\\delta$ in\n$\\ensuremath{\\mathbb Z}[i]$,\n\\[  a_r^*:=a_{r\\delta}\\chi_r(d), \\]\nand\n\\begin{equation*}\nB_4(Q,M):=\\sup\\limits_{(a_m)} \\|a_m \\|^{-2}\n\\sideset{}{^*}\\sum\\limits_{\\substack{M<m_1, m_2\\le 2M\\\\ (m_1m_2,2)=1 \\\\\n(m_1,m_2)=1}} a_{m_1}\\overline{a}_{m_2} \\sum\\limits_{n\\in \\ensuremath{\\mathbb Z}[i]}\nW\\left(\\frac{\\mathcal{N}(n)}{Q}\\right)\n\\chi_{m_1}(n)\\overline{\\chi}_{m_2}(n).\n\\end{equation*}\n\nMoreover, we define a norm $C_1(M,Q)$ dual to $B_1(Q,M)$ by\n\\begin{equation*}\nC_1(M,Q):=\\sup\\limits_{(b_{n})} \\| b_n \\|^{-2}\n\\sideset{}{^*}\\sum\\limits_{\\substack{M<m\\le 2M\\\\ (m,2)=1}} \\left|\n\\sideset{}{'}\\sum\\limits_{\\substack{n\\in \\ensuremath{\\mathbb Z}[i]\\\\ Q<\\mathcal{N}(n)\\le 2Q\\\\\nn\\equiv \\pm 1 \\bmod {(1+i)^3}}} b_{n} \\chi_n(m)\\right|^2.\n\\end{equation*}\nBy the duality principle, we have\n\\begin{equation} \\label{B1C1}\nB_1(Q,M)=C_1(M,Q).\n\\end{equation}\nFurthermore, we define a norm $C_2(M,Q)$ by extending the\nsummation over $m$ in the definition of $C_1(M,Q)$ to all integers\n$m$ with $M < m \\leq 2M$. Trivially, we have\n\\begin{equation} \\label{C12}\nC_1(M,Q)\\le C_2(M,Q).\n\\end{equation}\n\n\\subsection{Comparison of the norms} \\label{section:comparison}\n\nFor the proof of our Theorem \\ref{quarticlargesieve}, we need the\nfollowing lemma on the norms defined in the previous section.\n\n\\begin{lemma} \\label{normlemma} Let $Q,M\\ge 1$ and $C$ be a sufficiently large positive constant. Then we have the following inequalities:\n\\begin{equation}\n\\label{C2e1} C_2(M,Q) \\ll (QM)^{\\varepsilon}\\left(M +\nQ^{7\/4}\\right);\n\\end{equation}\n\\begin{equation} \\label{C22}\nC_2(M,Q)\\ll\nM^{\\varepsilon}Q^{1-1\/v}\\sum\\limits_{j=0}^{v-1}C_2(2^jM^v,Q)^{1\/v},\n\\quad \\text{ for each fixed positive integer $v$};\n\\end{equation}\n\\begin{equation} \\label{B11'}\nB_1(Q_1,M)\\ll B_1(Q_2,M), \\quad \\text{ if $Q_1,M\\ge 1$ and $Q_2\\ge\nCQ_1\\log (2Q_1M)$};\n\\end{equation}\n\\begin{equation} \\label{B21}\nB_2(Q,M) \\ll (\\log{2Q})^3 Q^{1\/2}X^{-1\/2} B_1(XQ^{\\varepsilon},M),\n\\quad \\text{for some $X$ with $1\\le X\\le Q$};\n\\end{equation}\n\\begin{equation} \\label{B32}\nB_3(Q,M) \\ll (\\log 2Q)^3 Q^{1\/2}X^{-1\/2} B_2(XQ^{\\varepsilon},M),\n\\quad \\text{for some $X$ with $1\\le X\\le Q$};\n\\end{equation}\n\\begin{equation} \\label{B34}\nB_3(Q,M)\\ll M^{\\varepsilon}\nB_4\\left(\\frac{Q}{\\Delta_1},\\frac{M}{\\Delta_2}\\right), \\quad\n\\text{for some $\\Delta_1,\\Delta_2\\in \\mathbb{N}$ with\n$\\Delta_2^2\\ge \\Delta_1$};\n\\end{equation}\n\\begin{equation} \\label{B43}\nB_4(Q,M)\\ll Q+QM^{\\varepsilon-2} \\max\\left\\{B_3(K,M) : K\\le\nM^4Q^{-1}\\right\\} + Q^{-1}M^{6+\\varepsilon} \\sum\\limits_{K> M^4\/Q}\nK^{-2-\\varepsilon} B_3(K,M),\n\\end{equation}\nwhere the sum over $K$ in \\eqref{B43} runs over powers of $2$.\n\\end{lemma}\n\nSince the proofs of \\eqref{C22}-\\eqref{B43} are essentially the\nsame as those of (31)-(36) of Lemma 4.1 in \\cite{B&Y}, we omit\nthe proofs here. \\newline\n\nWe note that it follows from \\eqref{B1C1}-\\eqref{C22}, that we have\n\\begin{equation} \\label{C2egen}\nB_1(Q,M) \\ll (QM)^{\\varepsilon}\\left(Q^{1-1\/v} M +\nQ^{1+3\/(4v)}\\right)\n\\end{equation}\nfor any $v\\in \\ensuremath{\\mathbb N}$.\n\n\n\\subsection{Estimating $C_2$} \\label{section:C2}\n\n  In this section we prove \\eqref{C2e1}. Recall $C_2(M,Q)$\nis the norm of the sum\n\\begin{equation}\n\\label{eq:C2def} \\sum\\limits_{M<m\\le 2M} \\left|\n\\sideset{}{'}\\sum\\limits_{\\substack{n\\in \\ensuremath{\\mathbb Z}[i]\\\\ Q<\\mathcal{N}(n)\\le 2Q\\\\\nn\\equiv \\pm 1 \\bmod {(1+i)^3}}}\n b_{n} \\chi_n(m)\\right|^2,\n\\end{equation}\nwhere the apostrophe indicates that $n$ is square-free and has no\nrational prime divisor. \\newline\n\nThe sum in \\eqref{eq:C2def} is obviously bounded by\n\\begin{equation}\n\\label{eq:C2defW}\n \\ll \\sum\\limits_{m\\in \\ensuremath{\\mathbb Z}}\nW\\left(\\frac{m}{M}\\right) \\left| \\sideset{}{'}\\sum\\limits_{n} b_{n}\n\\chi_n(m)\\right|^2,\n\\end{equation}\nwhere the weight function $W$ is defined as in \\eqref{W}.\nExpanding out the sum in \\eqref{eq:C2defW} we get\n\\begin{equation*}\n\\sideset{}{'}\\sum \\limits_{\\substack{n_1,n_2\\in \\ensuremath{\\mathbb Z}[i]\\\\\nQ<\\mathcal{N}(n_1),\\mathcal{N}(n_2)\\le 2Q\\\\ n_1,n_2\\equiv \\pm 1 \\bmod {(1+i)^3}}}\nb_{n_1}\\overline{b}_{n_2} \\sum\\limits_{m\\in \\ensuremath{\\mathbb Z}}\nW\\left(\\frac{m}{M}\\right) \\chi_{n_1}\\overline{\\chi}_{n_2}(m).\n\\end{equation*}\nNow we extract the greatest common divisor $\\Delta$ of $n_1$ and\n$n_2$, getting\n\\begin{equation*}\n\\sideset{}{'}\\sum_{\\substack{\\mathcal{N}(\\Delta) \\leq 2Q \\\\ \\Delta \\equiv 1\n\\bmod{{(1+i)^3}}}}\\ \\sideset{}{'}\\sum \\limits_{\\substack{n_1,n_2\\in\n\\ensuremath{\\mathbb Z}[i]\\\\ \\frac{Q}{\\mathcal{N}(\\Delta)}<\\mathcal{N}(n_1),\\mathcal{N}(n_2)\\le\n\\frac{2Q}{\\mathcal{N}(\\Delta)} \\\\ n_1,n_2\\equiv 1 \\bmod {(1+i)^3} \\\\\n(n_1,n_2)=1 \\\\ (n_1 n_2, \\Delta \\overline{\\Delta}) = 1 }} b_{n_1\n\\Delta}\\overline{b}_{n_2 \\Delta} \\sum\\limits_{\\substack{m\\in \\ensuremath{\\mathbb Z}\n\\\\ (m, \\mathcal{N}(\\Delta)) = 1}} W\\left(\\frac{m}{M}\\right)\n\\chi_{n_1}\\overline{\\chi}_{n_2}(m).\n\\end{equation*}\nIt is easy to see that there is a one-to-one correspondence between the pairs $(n_1,n_2)$ and $(n'_1\\Delta,n'_2 \\Delta)$ with $\\Delta \\equiv 1$, $n_1 \\equiv n'_1$ and $n_2 \\equiv n'_2 \\bmod{(1+i)^3}$.  We write $\\delta = (n_1, \\bar{n}_2)$ and change variables\nvia $n_1 \\rightarrow \\delta n_1$, $n_2 \\rightarrow\n\\overline{\\delta} n_2$ to get\n\\begin{equation} \\label{eq:prePoisson}\n\\begin{split}\n\\sideset{}{'}\\sum_{\\substack{\\mathcal{N}(\\Delta) \\leq 2Q \\\\\n\\Delta \\equiv 1 \\bmod{{(1+i)^3}}}}\\\n\\sideset{}{'}\\sum_{\\substack{\\mathcal{N}(\\delta)\n\\leq \\frac{2Q}{\\mathcal{N}(\\Delta)} \\\\ \\delta \\equiv 1 \\bmod{{(1+i)^3}} \\\\\n(\\mathcal{N}(\\delta), \\mathcal{N}(\\Delta)) = 1}} & \\\n\\sideset{}{'}\\sum\\limits_{\\substack{n_1,n_2\\in \\ensuremath{\\mathbb Z}[i]\\\\\n\\frac{Q}{\\mathcal{N}(\\delta \\Delta)}<\\mathcal{N}(n_1),\\mathcal{N}(n_2)\\le \\frac{2Q}{\\mathcal{N}(\\delta\n\\Delta)} \\\\ n_1,n_2\\equiv \\pm 1 \\bmod {(1+i)^3} \\\\ (\\mathcal{N}(n_1),\n\\mathcal{N}(n_2)) = 1 \\\\ (n_1 n_2, \\delta \\overline{\\delta} \\Delta\n\\overline{\\Delta}) = 1 } } b_{n_1 \\Delta \\delta}\\overline{b}_{n_2\n\\Delta \\overline{\\delta}}\n\\\\\n& \\times \\sum\\limits_{\\substack{m\\in \\ensuremath{\\mathbb Z} \\\\ (m, \\mathcal{N}(\\Delta)) = 1}}\nW\\left(\\frac{m}{M}\\right)\n\\chi_{n_1}\\overline{\\chi}_{n_2}\\overline{\\chi}_{\\delta}^2(m),\n\\end{split}\n\\end{equation}\nwhere we use that for $m \\in \\ensuremath{\\mathbb Z}$,\n$\\chi_{\\delta}\\overline{\\chi}_{\\overline{\\delta}}(m)=\\chi_{\\delta}^2(m)=\n\\overline{\\chi}_{\\delta}^2(m)$. Next we remove the coprimality\ncondition in the sum over $m$ by the M\\\"obius function, getting\n\\begin{equation*}\n\\sum\\limits_{\\substack{m\\in \\ensuremath{\\mathbb Z} \\\\ (m, \\mathcal{N}(\\Delta)) = 1}}\nW\\left(\\frac{m}{M}\\right)\n\\chi_{n_1}\\overline{\\chi}_{n_2}\\overline{\\chi}_{\\delta}^2(m) =\n\\sum_{\\ell | \\mathcal{N}(\\Delta)} \\mu(\\ell)\n\\chi_{n_1}\\overline{\\chi}_{n_2}\\overline{\\chi}_{\\delta}^2(\\ell)\n\\sum\\limits_{\\substack{m\\in \\ensuremath{\\mathbb Z} }} W\\left(\\frac{m}{M\/\\ell}\\right)\n\\chi_{n_1}\\overline{\\chi}_{n_2}\\overline{\\chi}_{\\delta}^2(m),\n\\end{equation*}\nwhich by the Poisson summation formula is\n\\begin{equation} \\label{afterpoisson}\n\\begin{split}\n\\sum_{\\ell | \\mathcal{N}(\\Delta)} \\mu(\\ell)\n\\chi_{n_1}\\overline{\\chi}_{n_2}\\overline{\\chi}_{\\delta}^2(\\ell)\n& \\frac{M}{\\ell \\mathcal{N}(n_1 n_2 \\delta)} \\sum\\limits_{\\substack{h\\in \\ensuremath{\\mathbb Z}}}\n\\widehat{W}\\left(\\frac{hM}{\\ell \\mathcal{N}(n_1 n_2 \\delta)}\\right)\n\\\\\n& \\times \\sum_{r \\bmod{\\mathcal{N}(n_1 n_2 \\delta)}}\n\\chi_{n_1}\\overline{\\chi}_{n_2}\\overline{\\chi}_{\\delta}^2(r)\ne\\left(\\frac{rh}{\\mathcal{N}(n_1 n_2 \\delta)}\\right),\n\\end{split}\n\\end{equation}\n  where\n\\begin{align*}\n  \\widehat{W}(x)=\\int\\limits^{\\infty}_{-\\infty}W(y)e(-xy)dy.\n\\end{align*}\n\nWhen $h=0$, the expression in \\eqref{afterpoisson} vanishes unless\n$n_1 = n_2 = \\delta =1$. Hence, the contribution of $h=0$ to\n\\eqref{eq:prePoisson} is\n\\begin{equation} \\label{h0cont}\n\\ll MQ^{\\varepsilon} \\sideset{}{'}\\sum_{\\substack{Q<\\mathcal{N}(\\Delta) \\leq 2Q \\\\\n\\Delta \\equiv 1 \\bmod{{(1+i)^3}}}} |b_{\\Delta}|^2 \\ll M\nQ^{\\varepsilon}\\| b_n \\|^2.\n\\end{equation}\n\nIn the sequel, we assume that $h\\not=0$. The sum over $r$ in\n\\eqref{afterpoisson} can be computed by writing $r = r_1 \\mathcal{N}(n_2\n\\delta) + r_2 \\mathcal{N}(n_1 \\delta) + r_3 \\mathcal{N}(n_1 n_2)$ to get\n\\begin{equation} \\label{gauss}\n\\begin{split}\n&\\sum_{r \\bmod{\\mathcal{N}(n_1n_2\\delta)}}  \\chi_{n_1}\\overline{\\chi}_{n_2}\\overline{\\chi}_{\\delta}^2(r) e\\left(\\frac{rh}{\\mathcal{N}(n_1 n_2 \\delta)}\\right)\\\\\n& = \\sum_{r_1 \\bmod{\\mathcal{N}(n_1)}} \\chi_{n_1}(r_1\\mathcal{N}(n_2\\delta))\ne\\left(\\frac{r_1h}{\\mathcal{N}(n_1)}\\right) \\sum_{r_2 \\bmod{\\mathcal{N}(n_2)}}\n\\overline{\\chi}_{n_2}(r_2\\mathcal{N}(n_1\\delta))\ne\\left(\\frac{r_2h}{\\mathcal{N}(n_2)}\\right) \\\\\n&  \\hspace*{3cm} \\times \\sum_{r_3\n\\bmod{\\mathcal{N}(\\delta)}}  \\overline{\\chi}_{\\delta}^2(r_3\\mathcal{N}(n_1n_2))\ne\\left(\\frac{r_3h}{\\mathcal{N}(\\delta)}\\right) \\\\\n&= \\overline{\\chi}_{n_1}(h) \\chi_{n_2}\\chi^2_{\\delta}(-h)\n\\chi_{n_1}(\\mathcal{N}(n_2 \\delta)) \\overline{\\chi}_{n_2}(\\mathcal{N}(n_1 \\delta))\n\\overline{\\chi}_{\\delta}^2(\\mathcal{N}(n_1n_2)) \\tau(\\chi_{n_1})\n\\overline{\\tau(\\chi_{n_2})\\tau(\\chi^2_{\\delta})},\n\\end{split}\n\\end{equation}\n   where $\\tau (\\chi)$ is defined as in \\eqref{tau}. \\newline\n\nUsing quartic reciprocity and the identity\n\\begin{equation*}\n\\left(\\frac{m}{n}\\right)_4 =\n\\overline{\\left(\\frac{\\overline{m}}{\\overline{n}}\\right)}_4\n\\end{equation*}\nfollowing from the definition of the quartic residue symbol, we\nget the identities\n\\begin{align*}\n\\chi_{n}(\\mathcal{N}(m))\\overline{\\chi}_{m}(\\mathcal{N}(n))&=\\left(\\frac{\\mathcal{N}(m)}{n}\\right)_4\n\\overline{\\left(\\frac{\\mathcal{N}(n)}{m}\\right)}_4 =\n\\left(\\frac{m}{\\overline{n}}\\right)^2_4 = \\chi^2_{\\overline{n}}(m)\n\\\\\n\\chi_{n}(\\mathcal{N}(m))\\overline{\\chi}_{m}^2(\\mathcal{N}(n))&=\\left(\\frac{\\mathcal{N}(m)}{n}\\right)_4\n\\left(\\frac{\\mathcal{N}(n)}{m}\\right)^2_4 =\n\\left(\\frac{\\mathcal{N}(m)}{\\overline{n}}\\right)_4 =\n\\chi_{\\overline{n}}(\\mathcal{N}(m))\n\\end{align*}\nand\n\\begin{equation*}\n\\overline{\\chi}_{n}(\\mathcal{N}(m))\\overline{\\chi}_{m}^2(\\mathcal{N}(n))=\n\\overline{\\left(\\frac{\\mathcal{N}(m)}{n}\\right)}_4\n\\overline{\\left(\\frac{\\mathcal{N}(n)}{m}\\right)}^2_4 =\n\\left(\\frac{\\mathcal{N}(m)}{n}\\right)_4 = \\chi_{n}(\\mathcal{N}(m)),\n\\end{equation*}\nvalid for all $m,n\\in \\ensuremath{\\mathbb Z}[i]$ with $m, n \\equiv \\pm 1\n\\bmod{{(1+i)^3}}$. We use them to simplify the last line of\n\\eqref{gauss}, obtaining\n\\begin{equation*}\n  \\overline{\\chi}_{n_1}(h)\n\\chi_{n_2}\\chi^2_{\\delta}(-h)  \\left(\\frac{n^2_2\n\\mathcal{N}(\\delta)}{\\overline{n_1}}\\right)_4\n\\left(\\frac{\\mathcal{N}(\\delta)}{n_2}\\right)_4 \\tau(\\chi_{n_1})\n\\overline{\\tau(\\chi_{n_2})\\tau(\\chi^2_{\\delta})}.\n\\end{equation*}\n\nNow, changing $n_2\\rightarrow \\overline{n}_2$, the contribution of\n$h\\not=0$ to the sum in \\eqref{afterpoisson} takes the form\n\\begin{multline}\nS_W(M,Q)=M \\sideset{}{'}\\sum_{\\substack{\\mathcal{N}(\\Delta) \\leq 2Q \\\\ \\Delta \\equiv 1 \\bmod{{(1+i)^3}}}}\\ \\sideset{}{'}\\sum_{\\substack{\\mathcal{N}(\\delta) \\leq\n\\frac{2Q}{\\mathcal{N}(\\Delta)} \\\\ \\delta \\equiv 1 \\bmod{{(1+i)^3}} \\\\\n(\\mathcal{N}(\\delta), \\mathcal{N}(\\Delta)) =\n1}}\\frac{\\overline{\\tau(\\chi_{\\delta})}}{\\mathcal{N}(\\delta)} \\sum_{\\ell |\n\\mathcal{N}(\\Delta)} \\frac{\\mu(\\ell)}{\\ell} \\overline{\\chi}_{\\delta}^2(\\ell)\n\\sum_{h\\not=0}  \\chi^2_{\\delta}(h)\n\\\\\n\\times \\sideset{}{'}\\sum\\limits_{\\substack{n_1,n_2\\in \\ensuremath{\\mathbb Z}[i]\\\\\n\\frac{Q}{\\mathcal{N}(\\delta \\Delta)}<\\mathcal{N}(n_1),\\mathcal{N}(n_2)\\le \\frac{2Q}{\\mathcal{N}(\\delta\n\\Delta)} \\\\ n_1,n_2\\equiv \\pm 1 \\bmod {(1+i)^3} \\\\ (\\mathcal{N}(n_1),\n\\mathcal{N}(n_2)) = 1 \\\\ (n_1 n_2, \\delta \\overline{\\delta} \\Delta\n\\overline{\\Delta}) = 1 } } \\widehat{W}\\left(\\frac{hM}{\\ell \\mathcal{N}(n_1 n_2\n\\delta)}\\right) c_{\\Delta,\\delta,\\ell,h,n_1}\nc_{\\Delta,\\delta,\\ell,h,n_2}'\n\\left(\\frac{n_1}{n_2}\\right)^2_4,\\nonumber\n\\end{multline}\nwhere\n\\[\nc_{\\Delta,\\delta,\\ell,h,n}:=\\chi_{n}(\\ell)\\overline{\\chi}_n(h)\n\\left(\\frac{\\mathcal{N}(\\delta)}{\\overline{n}}\\right)_4\\frac{\\tau(\\chi_n)}{\\mathcal{N}(n)}\nb_{n\\Delta\\delta} \\; \\; \\; \\mbox{and} \\; \\; \\;\nc_{\\Delta,\\delta,\\ell,h,n}':= \\chi_{n}(\\ell)\\overline{\\chi}_n(h)\n\\left(\\frac{\\mathcal{N}(\\delta)}{\\overline{n}}\\right)_4\\frac{\\tau(\\chi_n)}{\\mathcal{N}(n)}\n\\overline{b}_{\\overline{n}\\Delta\\overline{\\delta}}.\n\\]\n We now estimate the sum over $n_1$ and $n_2$ directly using\n\\eqref{eq:quartic}. We denote the inner sum in the definition of\n$S_W(M,Q)$ above to be $U(\\Delta, \\delta, \\ell, h)$ so that\n\\begin{align*}\n  U(\\Delta, \\delta, \\ell,\nh)= \\sideset{}{'}\\sum\\limits_{\\substack{n_1,n_2\\in \\ensuremath{\\mathbb Z}[i]\\\\\n\\frac{Q}{\\mathcal{N}(\\delta \\Delta)}<\\mathcal{N}(n_1),\\mathcal{N}(n_2)\\le \\frac{2Q}{\\mathcal{N}(\\delta\n\\Delta)} \\\\ n_1,n_2\\equiv \\pm 1 \\bmod {(1+i)^3} \\\\ (\\mathcal{N}(n_1),\n\\mathcal{N}(n_2)) = 1 \\\\ (n_1 n_2, \\delta \\overline{\\delta} \\Delta\n\\overline{\\Delta}) = 1 } } \\widehat{W}\\left(\\frac{hM}{\\ell \\mathcal{N}(n_1 n_2\n\\delta)}\\right) c_{\\Delta,\\delta,\\ell,h,n_1}\nc_{\\Delta,\\delta,\\ell,h,n_2}' \\left(\\frac{n_1}{n_2}\\right)^2_4.\n\\end{align*}\n  To separate the variables $n_1,n_2$, we remove the coprimality\ncondition $(\\mathcal{N}(n_1),\\mathcal{N}(n_2))=1$. Because of the\npresence of $(\\frac{n_1}{n_2})_4$, we may assume that $n_1$ and\n$n_2$ are coprime.  Thus $\\mathcal{N}(n_1)$ and $\\mathcal{N}(n_2)$\nmust also be coprime unless $n_1$ has a factor in common with\n$\\overline{n}_2$. We proceed to detect this latter condition using\nthe Mobius function in the standard way to obtain\n\n\\begin{align*}\n  U(\\Delta, \\delta, \\ell,\nh)&= \\sideset{}{'}\\sum_{\\substack{e \\in \\ensuremath{\\mathbb Z}[i] \\\\ e \\equiv 1 \\bmod {(1+i)^3} \\\\ (\\mathcal{N}(e), \\mathcal{N}(\\delta\\Delta))=1 }} \\mu_{[i]}(e) \\sideset{}{'}\\sum\\limits_{\\substack{n_1,n_2\\in \\ensuremath{\\mathbb Z}[i]\\\\\n\\frac{Q}{\\mathcal{N}(\\delta\n\\Delta)}<\\mathcal{N}(n_1),\\mathcal{N}(n_2)\\le\n\\frac{2Q}{\\mathcal{N}(\\delta \\Delta)} \\\\ n_1,n_2\\equiv \\pm 1 \\bmod\n{(1+i)^3} \\\\ e|n_1, \\overline{e}| n_2 \\\\ (n_1 n_2, \\delta\n\\overline{\\delta} \\Delta \\overline{\\Delta}) = 1 } }\n\\widehat{W}\\left(\\frac{hM}{\\ell \\mathcal{N}(n_1 n_2\n\\delta)}\\right) c_{\\Delta,\\delta,\\ell,h,n_1}\nc_{\\Delta,\\delta,\\ell,h,n_2}' \\left(\\frac{n_1}{n_2}\\right)^2_4 \\\\\n&= \\sideset{}{'}\\sum_{\\substack{e \\in \\ensuremath{\\mathbb Z}[i] \\\\ e \\equiv 1 \\bmod {(1+i)^3}\n\\\\ \\mathcal{N}(e) \\leq \\frac{2Q}{\\mathcal{N}(\\delta \\Delta)} \\\\\n(\\mathcal{N}(e), \\mathcal{N}(\\delta\\Delta))=1  }}\n\\mu_{[i]}(e)\\chi_{\\mathcal{N}(e)}(\\ell)\\overline{\\chi}_{\\mathcal{N}(e)}(h)\\left(\\frac{\\mathcal{N}(\\delta)}{\\mathcal{N}(e)}\\right)_4\\frac{\\tau(\\chi_e)}{\\mathcal{N}(e)}\n\\frac{\\tau(\\chi_{\\overline{e}})}{\\mathcal{N}(\\overline{e})}\\left(\\frac{e}{\\overline{e}}\\right)^2_4 \\\\\n& \\hspace*{2cm} \\times \\sideset{}{'}\\sum\\limits_{\\substack{n_1,n_2\\in \\ensuremath{\\mathbb Z}[i]\\\\\n\\frac{Q}{\\mathcal{N}(e\\delta \\Delta)}<\\mathcal{N}(n_1),\\mathcal{N}(n_2)\\le \\frac{2Q}{\\mathcal{N}(e\\delta\n\\Delta)} \\\\ n_1,n_2\\equiv \\pm 1 \\bmod {(1+i)^3} \\\\  (n_1 n_2,\ne \\overline{e}\\delta \\overline{\\delta} \\Delta \\overline{\\Delta}) =\n1 } } \\widehat{W}\\left(\\frac{hM}{\\ell \\mathcal{N}(n_1 n_2 e\n\\overline{e}\\delta)}\\right) c_{\\Delta,\\delta,\\ell,h,e,n_1}\nc'_{\\Delta,\\delta,\\ell,h,e, n_2} \\left(\\frac{n_1}{n_2}\\right)^2_4 .\n\\end{align*}\n   where\n\\[ c_{\\Delta,\\delta,\\ell,h,e,n} :=\\chi_{n}(\\ell)\\overline{\\chi}_n(h)\n\\left(\\frac{\\mathcal{N}(\\delta)}{\\overline{n}}\\right)_4\\left(\\frac{\\mathcal{N}(e)}{n}\\right)_4\\left(\\frac{\\mathcal{N}(n)}{e}\\right)_4\n\\left(\\frac{n_1}{\\overline{e}}\\right)^2_4\\frac{\\tau(\\chi_n)}{\\mathcal{N}(n)}\nb_{ne\\Delta\\delta} \\]\nand\n\\[ c'_{\\Delta,\\delta,\\ell,h,e,n} :=\\chi_{n}(\\ell)\\overline{\\chi}_n(h)\n\\left(\\frac{\\mathcal{N}(\\delta)}{\\overline{n}}\\right)_4\\left(\\frac{\\mathcal{N}(e)}{n}\\right)_4\\left(\\frac{\\mathcal{N}(n)}{\\overline{e}}\\right)_4\\left(\\frac{e}{n_2}\\right)^2_4\\frac{\\tau(\\chi_n)}{\\mathcal{N}(n)}\n\\overline{b}_{\\overline{n}e\\Delta\\overline{\\delta}}. \\]\n Next observe that we may freely truncate the sum over $h$ for\n\\begin{equation*}\n|h| \\le \\frac{Q^2 \\ell}{\\mathcal{N}(\\delta)\\mathcal{N}(\\Delta)^2M} (QM)^{\\varepsilon}=:H\n\\end{equation*}\nsince $\\widehat{W}$ has rapid decay. More precisely, if we let $S_W\n(M,Q) = S'_W(M,Q) + E$ where $S'_W(M,Q)$ is the contribution to\n$S_W(M,Q)$ from $0 < |h| \\leq H$, then $E \\ll\n(MQ)^{-100} \\| b \\|^2$. \\newline\n\n  It remains to bound $S'_W(M,Q)$ and we have\n\\begin{equation*}\n\\begin{split}\nS'_W(M,Q) \\ll M \\sideset{}{'}\\sum_{\\substack{\\mathcal{N}(\\Delta) \\leq 2Q \\\\ \\Delta\n\\equiv 1 \\bmod{{(1+i)^3}}}}\\\n\\sideset{}{'}\\sum_{\\substack{\\mathcal{N}(\\delta) \\leq\n\\frac{2Q}{\\mathcal{N}(\\Delta)} \\\\ \\delta \\equiv 1 \\bmod{{(1+i)^3}} \\\\\n(\\mathcal{N}(\\delta), \\mathcal{N}(\\Delta)) = 1}} & \\frac{1}{(\\mathcal{N}(\\delta))^{1\/2}} \\sum_{\\ell |\n\\mathcal{N}(\\Delta)} \\frac{1}{\\ell} \\\\\n& \\times \\sideset{}{'}\\sum_{\\substack{e \\in \\ensuremath{\\mathbb Z}[i] \\\\ e\n\\equiv 1 \\bmod {(1+i)^3}  \\\\ \\mathcal{N}(e) \\leq \\frac{2Q}{\\mathcal{N}(\\delta\n\\Delta)}\\\\ (\\mathcal{N}(e), \\mathcal{N}(\\delta\\Delta))=1  }}\\frac {1}{\\mathcal{N}(e)} \\sum_{0<\n|h| \\leq H} \\left| U'(\\Delta, \\delta, \\ell, e, h) \\right|,\n\\end{split}\n\\end{equation*}\n  where\n\\begin{align*}\n  U'(\\Delta, \\delta, \\ell, e,\nh)=  \\sideset{}{'}\\sum\\limits_{\\substack{n_1,n_2\\in \\ensuremath{\\mathbb Z}[i]\\\\\n\\frac{Q}{\\mathcal{N}(e\\delta \\Delta)}<\\mathcal{N}(n_1),\\mathcal{N}(n_2)\\le \\frac{2Q}{\\mathcal{N}(e\\delta\n\\Delta)} \\\\ n_1,n_2\\equiv \\pm 1 \\bmod {(1+i)^3} \\\\  (n_1 n_2,\ne \\overline{e}\\delta \\overline{\\delta} \\Delta \\overline{\\Delta}) =\n1 } } \\widehat{W}\\left(\\frac{hM}{\\ell \\mathcal{N}(n_1 n_2 e\n\\overline{e}\\delta)}\\right) c_{\\Delta,\\delta,\\ell,h,e,n_1}\nc'_{\\Delta,\\delta,\\ell,h,e, n_2} \\left(\\frac{n_1}{n_2}\\right)^2_4 .\n\\end{align*}\n    We  now remove the weight $\\widehat{W}$ in $U'(\\Delta, \\delta, \\ell, e,\nh)$ by applying Lemma \\ref{lem8} to the function $\\rho(x) =\n\\widehat{W}(x)$, which satisfies the conditions of that lemma, as one sees by repeated integration by parts.  We may assume $h>0$ here, since the contribution of the negative $h$'s can be treated similarly and satisfies the same bound.  We use $\\sigma = \\varepsilon$\nto see that\n\\begin{align*}\n    U'(\\Delta, \\delta, \\ell, e,\nh) & \\ll_{\\varepsilon}\n\\left(\\frac{\\ell \\mathcal{N}(e\\overline{e}\\Delta)}{hM}\\right)^{\\varepsilon}\\int\\limits^{\\infty}_{-\\infty}|\\rho_{+}(\\varepsilon+it)||V(\\varepsilon+it)|\n\\mathrm{d} t,\n\\end{align*}\n  where\n\\begin{align*}\n   V(s)= \\sideset{}{'}\\sum\\limits_{\\substack{n_1,n_2\\in \\ensuremath{\\mathbb Z}[i]\\\\\n\\frac{Q}{\\mathcal{N}(e\\delta \\Delta)}<\\mathcal{N}(n_1),\\mathcal{N}(n_2)\\le \\frac{2Q}{\\mathcal{N}(e\\delta\n\\Delta)} \\\\ n_1,n_2\\equiv \\pm 1 \\bmod {(1+i)^3} \\\\  (n_1 n_2,\ne \\overline{e}\\delta \\overline{\\delta} \\Delta \\overline{\\Delta}) =\n1 } } d_{n_1}d'_{n_2}\\left(\\frac{n_1}{n_2}\\right)^2_4,\n\\end{align*}\n   with\n\\begin{equation*}\n    d_n=c_{\\Delta,\\delta,\\ell,h,e,n}\\mathcal{N}(n)^{s} \\; \\; \\;  \\mbox{and} \\; \\; \\; d'_n=c'_{\\Delta,\\delta,\\ell,h,e,n}\\mathcal{N}(n)^{s} .\n\\end{equation*}\n   Note that $d_{n_1}$ and $d_{n_2}'$ depend on\n$\\Delta,\\delta, \\ell, h, e, n$ and $s$, and\n\\begin{equation*}\n|d_n| \\ll \\left(\\frac{\\mathcal{N}(e\\delta\\Delta)}{Q}\\right)^{1\/2-\\varepsilon}\n\\left| b_{ne\\Delta\\delta} \\right|, \\quad |d_n'| \\ll\n\\left(\\frac{\\mathcal{N}(e\\delta\\Delta)}{Q}\\right)^{1\/2-\\varepsilon}\n\\left| b_{\\overline{n}e\\Delta\\overline{\\delta}} \\right|.\n\\end{equation*}\n Now, using the Cauchy-Schwarz inequality and the estimate\n\\eqref{eq:quartic} upon noting that this estimate remains valid if\nthe summation conditions $m,n\\equiv 1 \\bmod {(1+i)^3}$ therein are\nreplaced by $m,n \\equiv \\pm 1 \\bmod {(1+i)^3}$ and\n$\\left(\\frac{n}{m}\\right)_4$ replaced by\n$\\left(\\frac{n_1}{n_2}\\right)^2_4$, we bound $V(\\varepsilon+it)$ by\n\\begin{equation} \\label{square}\n\\begin{split}\n& \\left| V(\\varepsilon+it) \\right|^2 \\\\\n&\\ll \\sideset{}{'}\\sum\\limits_{\\substack{n_1\\in \\ensuremath{\\mathbb Z}[i]\\\\\n\\frac{Q}{\\mathcal{N}(e\\delta \\Delta)}<\\mathcal{N}(n_1)\\le \\frac{2Q}{\\mathcal{N}(e\\delta \\Delta)}\n\\\\ n_1 \\equiv \\pm 1 \\bmod {(1+i)^3} \\\\  (n_1, e\n\\overline{e}\\delta \\overline{\\delta} \\Delta \\overline{\\Delta}) = 1\n} } |d_{n_1}|^2 \\;\\;\\;\\; \\times\n\\sideset{}{'}\\sum\\limits_{\\substack{n_1\\in \\ensuremath{\\mathbb Z}[i]\\\\\n\\frac{Q}{\\mathcal{N}(e\\delta \\Delta)}<\\mathcal{N}(n_1)\\le \\frac{2Q}{\\mathcal{N}(e\\delta \\Delta)}\n\\\\ n_1 \\equiv \\pm 1 \\bmod {(1+i)^3} \\\\  (n_1, e\n\\overline{e}\\delta \\overline{\\delta} \\Delta \\overline{\\Delta}) = 1\n} } \\left|\\sideset{}{'}\\sum\\limits_{\\substack{n_2\\in \\ensuremath{\\mathbb Z}[i]\\\\\n\\frac{Q}{\\mathcal{N}(e\\delta \\Delta)}<\\mathcal{N}(n_2)\\le \\frac{2Q}{\\mathcal{N}(e\\delta \\Delta)}\n\\\\ n_2 \\equiv \\pm 1 \\bmod {(1+i)^3} \\\\  (n_2, e\n\\overline{e}\\delta \\overline{\\delta} \\Delta \\overline{\\Delta}) = 1\n} } d_{n_2}'\n\\left(\\frac{n_1}{n_2}\\right)^2_4 \\right|^2 \\\\\n&\\ll (QM)^{8\\varepsilon}\\left(\\frac{\\mathcal{N}(e\\delta\n\\Delta)}{Q}\\right)^{1\/4 -4\\varepsilon} \\left( \\sideset{}{'}\\sum_{\\substack{Q\/\\mathcal{N}(e)< \\mathcal{N}(n)\\leq 2Q\/\\mathcal{N}(e)\\\\\nn \\equiv \\pm 1 \\bmod {(1+i)^3}\\\\ (\\mathcal{N}(n), \\mathcal{N}(e))=1 }} |b_{en}|^2 \\right)^2.\n\\end{split}\n\\end{equation}\n   Since\n\\begin{align*}\n     \\int\\limits^{\\infty}_{-\\infty}|\\rho_{+}(\\sigma+it)| \\mathrm{d} t \\ll_{\\varepsilon} 1,\n\\end{align*}\n    we deduce that\n\\begin{equation} \\label{cont}\n\\begin{split}\nS'_W(M,Q) & \\ll  M(QM)^{7\\varepsilon}\n\\sideset{}{'}\\sum_{\\substack{\\mathcal{N}(\\Delta) \\leq 2Q \\\\ \\Delta \\equiv 1\n\\bmod{{(1+i)^3}}}}\\ \\sideset{}{'}\\sum_{\\substack{\\mathcal{N}(\\delta) \\leq\n\\frac{2Q}{\\mathcal{N}(\\Delta)} \\\\ \\delta \\equiv 1 \\bmod{{(1+i)^3}} \\\\\n(\\mathcal{N}(\\delta), \\mathcal{N}(\\Delta)) = 1}}\\frac{1}{(\\mathcal{N}(\\delta))^{1\/2}} \\sum_{\\ell |\n\\mathcal{N}(\\Delta)} \\frac{1}{\\ell} \\sideset{}{'}\\sum_{\\substack{e \\in \\ensuremath{\\mathbb Z}[i] \\\\ e\n\\equiv 1 \\bmod {(1+i)^3}  \\\\ \\mathcal{N}(e) \\leq \\frac{2Q}{\\mathcal{N}(\\delta\n\\Delta)}\\\\ (\\mathcal{N}(e), \\mathcal{N}(\\delta\\Delta))=1  }}\\frac {1}{\\mathcal{N}(e)}\n\\\\\n& \\hspace*{2cm} \\times \\sum_{0< |h| \\leq H} \\left(\\frac{\\mathcal{N}(e\\delta\n\\Delta)}{Q}\\right)^{\\frac14 -2\\varepsilon}\\sideset{}{'}\\sum_{\\substack{Q\/\\mathcal{N}(e)< \\mathcal{N}(n)\\leq 2Q\/\\mathcal{N}(e)\\\\\nn \\equiv \\pm 1 \\bmod {(1+i)^3}\\\\ (\\mathcal{N}(n), \\mathcal{N}(e))=1 }} |b_{en}|^2 \\\\\n& \\ll Q^{7\/4+2\\varepsilon}(QM)^{8\\varepsilon}\n\\sideset{}{'}\\sum_{\\substack{\\mathcal{N}(\\Delta) \\leq 2Q \\\\ \\Delta \\equiv 1\n\\bmod{{(1+i)^3}}}}\\frac {1}{\\mathcal{N}(\\Delta)^{7\/4+2\\varepsilon}}\n\\sideset{}{'}\\sum_{\\substack{\\mathcal{N}(\\delta) \\leq\n\\frac{2Q}{\\mathcal{N}(\\Delta)} \\\\ \\delta \\equiv 1 \\bmod{{(1+i)^3}} \\\\\n(\\mathcal{N}(\\delta), \\mathcal{N}(\\Delta)) = 1}}\\frac{1}{(\\mathcal{N}(\\delta))^{5\/4+2\\varepsilon}}\n\\sum_{\\ell | \\mathcal{N}(\\Delta)}1 \\\\\n& \\hspace*{2cm} \\times \\sideset{}{'}\\sum_{\\substack{e \\in \\ensuremath{\\mathbb Z}[i]\n\\\\ e \\equiv 1 \\bmod {(1+i)^3}  \\\\ \\mathcal{N}(e) \\leq \\frac{2Q}{\\mathcal{N}(\\delta\n\\Delta)}\\\\ (\\mathcal{N}(e), \\mathcal{N}(\\delta\\Delta))=1  }}\\frac\n{1}{\\mathcal{N}(e)^{3\/4+2\\varepsilon}}\n\\sideset{}{'}\\sum_{\\substack{Q\/\\mathcal{N}(e)< \\mathcal{N}(n)\\leq 2Q\/\\mathcal{N}(e)\\\\\nn \\equiv \\pm 1 \\bmod {(1+i)^3}\\\\ (\\mathcal{N}(n), \\mathcal{N}(e))=1 }} |b_{en}|^2 \\\\\n& \\ll Q^{7\/4+2\\varepsilon}(QM)^{8\\varepsilon}\\sideset{}{'}\\sum_{\\substack{Q< \\mathcal{N}(n)\\leq 2Q\\\\\nn \\equiv \\pm 1 \\bmod {(1+i)^3}}} |b_{n}|^2\n\\sideset{}{'}\\sum_{\\substack{e \\in \\ensuremath{\\mathbb Z}[i]\n\\\\ e \\equiv 1 \\bmod {(1+i)^3}  \\\\  e|n  }}\\frac\n{1}{\\mathcal{N}(e)^{3\/4+2\\varepsilon}} \\\\\n& \\ll Q^{7\/4+2\\varepsilon}(QM)^{9\\varepsilon}\\sideset{}{'}\\sum_{\\substack{Q< \\mathcal{N}(n)\\leq 2Q\\\\\nn \\equiv \\pm 1 \\bmod {(1+i)^3}}} |b_{n}|^2 .\n\\end{split}\n\\end{equation}\n\n  Combining \\eqref{h0cont} and \\eqref{cont}, we deduce that\n\\eqref{eq:prePoisson} and hence \\eqref{eq:C2def} is bounded by\n\\begin{equation*}\n\\ll (QM)^{\\varepsilon} \\left(M+Q^{7\/4}\\right) \\| b_{n} \\|^2\n\\end{equation*}\nwhich implies the desired bound \\eqref{C2e1}.\n\n\\section{Completion of the proof of Theorem \\ref{quarticlargesieve}}\nWe start with \\eqref{C2egen} with any $v\\ge 2$ (as one checks\neasily that $v=1$ does not lead to any improvement) as an initial\nestimate. From \\eqref{B21} and \\eqref{C2egen}, it follows that\n\\begin{equation*}\nB_2(Q,M) \\ll (QM)^{\\varepsilon} Q^{1\/2}X^{-1\/2}\n(X^{1+3\/(4v)}+X^{1-1\/v}M)\n\\end{equation*}\n   for a suitable $X$ with $1\\ll X\\ll Q$. The worst case is $X = Q$ which shows $B_2(Q,M)$ also satisfies\n   \\eqref{C2egen}. Repeating the argument, we have\n\\begin{equation*}\nB_3(Q,M)\\ll\n(QM)^{\\varepsilon}\\left(Q^{1+3\/(4v)}+Q^{1-1\/v}M\\right).\n\\end{equation*}\nCombining this with \\eqref{B43}, we obtain\n\\begin{eqnarray*}\nB_4(Q,M)&\\ll& Q+(QM)^{9\\varepsilon}QM^{-2} \\max\\left\\{K^{1+3\/(4v)}+K^{1-1\/v}M \\ :\\ K\\le M^4Q^{-1}\\right\\}\n\\\\ & & \\hspace*{1cm}+ (QM)^{9\\varepsilon}M^{6}Q^{-1} \\sum\\limits_{K\\ge M^4\/Q} K^{-2-\\varepsilon}(K^{1+3\/(4v)}+K^{1-1\/v}M)\\nonumber\\\\\n&\\ll&\nQ+(QM)^{10\\varepsilon}(Q^{-3\/(4v)}M^{2+3\/v}+Q^{1\/v}M^{3-4\/v}),\n\\end{eqnarray*}\nwhere the sum over $K$ runs over powers of $2$. From this and\n\\eqref{B34}, we deduce that\n\\begin{equation*}\nB_3(Q,M)\\ll \\frac{Q}{\\Delta_1}+(QM)^{\\varepsilon}\\left(\n\\left(\\frac{Q}{\\Delta_1}\\right)^{-3\/(4v)}\n\\left(\\frac{M}{\\Delta_2}\\right)^{2+3\/v}+\n\\left(\\frac{Q}{\\Delta_1}\\right)^{1\/v}\n\\left(\\frac{M}{\\Delta_2}\\right)^{3-4\/v}\\right)\n\\end{equation*}\nfor some positive integers $\\Delta_1$, $\\Delta_2$ with\n$\\Delta_2^2\\ge \\Delta_1$. From this and the trivial bound\n\\begin{equation*}\nB_1(Q,M)\\ll B_3(Q,M),\n\\end{equation*}\n we deduce that\n\\begin{equation}\n\\label{tem1} B_1(Q,M)\\ll\nQ+(QM)^{\\varepsilon}\\left(Q^{-3\/(4v)}M^{2+3\/v}+Q^{1\/v}M^{3-4\/v}\\right).\n\\end{equation}\nCombining \\eqref{tem1} with \\eqref{B11'}, we deduce that\n\\begin{equation} \\label{Re1}\nB_1(Q,M)\\ll\n(\\tilde{Q}M)^{\\varepsilon}\\left(\\tilde{Q}+\\tilde{Q}^{-3\/(4v)}M^{2+3\/v}+\\tilde{Q}^{1\/v}M^{3-4\/v}\\right)\n\\end{equation}\nif $\\tilde{Q}\\ge CQ\\log(2QM)$. We choose $\\tilde{Q}:=\\max\n(Q^{1+\\varepsilon}, M^{4-4v\/7}). $ Then \\eqref{Re1} implies that\n\\begin{equation} \\label{Re2}\nB_1(Q,M)\\ll (QM)^{\\varepsilon}\n\\left(Q+Q^{1\/v}M^{3-4\/v}+M^{17\/7}\\right).\n\\end{equation}\n\n  It's easy to see that the choice $v=2$ is optimal and a further cycle in the above process does not lead to an improvement of\nour result. Combining \\eqref{C2egen} with $v=1,2,3$ and\n\\eqref{Re2} with $v=2$, we obtain our final estimate\n\\begin{equation}\n\\label{final0} B_1(Q,M)\\ll\n(QM)^{\\varepsilon}\\min\\left\\{Q^{7\/4}+M, \\; Q^{11\/8}+Q^{1\/2}M, \\; Q^{5\/4}+Q^{2\/3}M,\\;\nQ+Q^{1\/2}M+M^{17\/7}\\right\\}.\n\\end{equation}\nwhich together with \\eqref{trans} (noting that the last expression in \\eqref{trans} is $\\ll B_1(Q,M)$ by the law of quartic reciprocity) implies Theorem \\ref{quarticlargesieve}. \\hfill $\\Box$ \\\\\n\nCalculating the right-hand side of \\eqref{final} for various\nranges of $Q$ and $M$, we obtain that it is bounded by\n\\begin{equation*}\n\\ll (QM)^{\\varepsilon} \\| a_m \\|^2\n\\cdot \\left\\{ \\begin{array}{llll} M &\\mbox{ if } Q\\le M^{4\/7},\\\\ \\\\ Q^{7\/4} &\\mbox{ if } M^{4\/7}<Q\\le M^{4\/5}, \\\\ \\\\\nQ^{1\/2}M &\\mbox{ if } M^{4\/5}<Q\\le M^{8\/7}, \\\\ \\\\\nQ^{11\/8} &\\mbox{ if } M^{8\/7}<Q\\le M^{24\/17}, \\\\ \\\\\nQ^{2\/3}M &\\mbox{ if } M^{24\/17}<Q\\le M^{12\/7}, \\\\ \\\\\nQ^{5\/4} &\\mbox{ if } M^{12\/7} <Q\\le M^{68\/35}, \\\\ \\\\\nM^{17\/7} &\\mbox{ if } M^{68\/35} <Q\\le M^{17\/7}, \\\\ \\\\\nQ & \\mbox{ if } M^{17\/7}<Q.\n\\end{array} \\right.\n\\end{equation*}\n  For convenience, we enclose a table displaying the estimates for $B_1(Q,M)$ that we get\nfor various ranges. This table should be read as follows. If the\nfractions $\\alpha$ and $\\beta$ are the $(n-1)$-th and $n$-th\nentries, respectively, in the first row, and the term $T$ is the\n$n$-th entry in the second row, then the estimate $B_1(Q,M) \\ll\n(QM)^{\\varepsilon}T$ holds in the range $M^{\\alpha} < Q \\leq\nM^{\\beta}$.\n\n\\begin{tabular}{|l||l|l|l|l|l|l|l|l|l|}\n\\hline Range & 4\/7 & 4\/5 & 8\/7 & 24\/17 & 12\/7 & 68\/35 & 17\/7 &  $\\infty$ \\\\\n\\hline\nBounds & $M$ & $Q^{7\/4}$ & $Q^{1\/2}$ & $Q^{11\/8}$ & $Q^{2\/3}$ & $Q^{5\/4}$ & $M^{17\/7}$ & $Q$  \\\\\n\\hline\n\\end{tabular} \\\\\n\nIt can be easily checked that \\eqref{C2egen} with $v\\ge 3$ does\nnot lead to an improvement of \\eqref{final0}. \\newline\n\n\\noindent{\\bf Acknowledgments.} During this work, P. G. was\nsupported by postdoctoral research fellowships at Nanyang\nTechnological University (NTU) and L. Z. by an AcRF Tier I Grant at NTU.  Moreover, L. Z. was on a visiting stay at {\\it Max-Planck-Institut f\\\"ur Mathematik} (MPIM) in Bonn and he wishes to thank MPIM for its financial support and hospitality during that pleasant and enjoyable period.  Finally, we thank the anonymous referee for his\/her many comments and suggestions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction:}\n\nEarly universe (standard big-bang) nucleosynthesis [SBBN] is\nregarded as a major success for the \nStandard Big Bang [SBB] Model. As presented, SBBN results look \nrather good indeed. The observed light element abundances are \ntaken to severely constrain cosmological and particle physics \nparameters. Deuterium, in particular, is regarded as\nan ideal ``baryometer'' for determining the baryon content of the \nuniverse \\cite{olive}. \nThis follows from the fact that deuterium is burned away whenever\nit is cycled through stars, and a belief, that there are no \nastrophysical sites (other than SBBN), capable of producing it \nin its observed abundance \\cite{eps}.\nThe  purpose of this  article is to admit caution in adhering to \nthis belief and to explore nucleosynthesis in an environment  \nradically different from the Standard.\n\n\tWhat would be the point of such an exercise ? \n\nIndeed, at the outset, drastic variations from SBBN may sound \npreposterous at this time. Confidence in SBBN stems primarily \nfrom $D$, $^7Li$ and $^4He$ measurements.  $D$ abundance is \nmeasured  in the solar wind, in interstellar clouds  and, more \nrecently, in the inter-galactic medium \\cite{geiss,hogan}. \nThe belief  that no realistic astrophysical process other than \nthe Big Bang  can  produce sufficient $D$ lends \nsupport to its primordial origin. Further, \n$^7Li$ measurement [$^7Li\/H \\sim 10^{-10}$] in Pop II stars \n\\cite{spite}  and the consensus \\cite{yang} over the primordial \nvalue for the $^4He$ ratio $Y_p \\ge 23.4 \\%$ \n(by mass) suggest that\nlight element abundances are consistent with SBBN over nine \norders of magnitude. This is  achieved by adjusting just\none parameter,  the baryon entropy ratio $\\eta$. Alternative \nmechanisms for $^7Li$ production that are \naccompanied by a co-production of $^6Li$ with a later depletion \nof $^7Li$ have fallen out of favour. The debate on depletion\nof $^7Li$ has been put to rest by the observation of $^6Li$ in a \nPop II star \\cite{smith}. Any depletion of $^7Li$ would have to \nbe accompanied by a  complete destruction of the much more \nfragile $^6Li$. Within the SBBN scenario therefore, one seeks to \naccount for the abundances of $^4He$, $D$, $^3He$ and $^7Li$ \ncosmologically, while $Be$, $B$ and $^6Li$ are generated  by \nspallation processes \\cite{olive1}.\n\n\nThese results, however, do meet with occasional skepticism\n[see eg. \\cite{primak} for problems with BBN]. \nObservation of $^6Li$, for example, requires unreasonable \nsuppression of astrophysical destruction of $^7Li$. On the \nother hand, the production of $^6Li$ would be accompanied by a \nsimultaneous production of $^7Li$ comparable to observed levels \n\\cite{rees}. This raises doubts about using observed $^7Li$ \nlevels as a benchmark to evaluate SBBN. \n\nFurther the best value of $^4He$ mass fraction, \nstatistically averaged and extrapolated\nto zero heavy element abundances, \nhovers around $.216 \\pm .006$ for Pop II objects \\cite{rana}.\nSuch low $^4He$ levels have also been reported     \nin several metal poor HII galaxies \\cite{pagel}. For example \nfor  SBS 0335-052 the reported value is $Y_p = 0.21 \\pm 0.01$ \n\\cite{ter}. Such small values for $^4He$ would not\nlead to any concordant value for $\\eta$ consistent with bounds on\n$^7Li$ and $D$. Of course, one could still explore a \nmulti-parameter non-minimal  SBBN instead of the minimal model \nthat just uses $\\eta$ for a single parameter fit. Non-vanishing \nneutrino chemical potentials have been proposed to be ``natural'' \nparameters for such a venture. These conclusions have been \ncriticized in \\cite{yang,ter} on grounds of reliance on statistical \nover-emphasis on a few metal-poor objects with a high enough \n$^4He$ abundance to save minimal SBBN. On the other hand, there \nare objects reported with abysmally low $^4He$ levels. This is \nalarming for  minimal SBBN. For example, levels of $^4He$ \ninferred for $\\mu$ Cassiopeiae A \\cite{ter} and from the emission\nlines of several quasars \\cite{peim} are as low as 5\\% and \n10 - 15\\% respectively. Such low levels would most definitely \nrule out SBBN. At present one excludes such objects from SBBN \nconsiderations on grounds of ``our lack of understanding''of the \nenvironments local to these objects. As a matter\nof fact,  one has to resort to specially contrived explanations to\naccount for low $Y_p$ values in quasars. Considering that a host \nof mechanisms for light element synthesis are discarded on grounds\nof requirement of special ``unnatural'' circumstances \\cite{eps}, \nit does not augur to have to resort to special explanations to \ncontend with low $^4He$ emission spectra. This comment ought to \nbe considered in the light of much emphasis that is \nlaid on emission lines from nebulae with low metal content \n\\cite{yang}. Quasars most certainly qualify for such candidates. \nInstead, one merely seems to  concentrate on classes of Pop II \nobjects and HII galaxies\nthat would oblige SBBN. Until the dependence of light \nelement abundance on sample and statistics is gotten rid  of and \n\/ or fully understood, one must not close one's eyes to \nalternatives.\n\n\nWe end our overview of the status of SBBN  with a few comments.\nFirstly the low metallicity that one sees in type II stars and \ninterstellar clouds poses a problem in SBBN. There is\nno object in the universe that has low abundance \n[metallicity] of heavier elements as is produced \nin SBB. One relies on some kind of re-processing, much later in \nthe history of the universe, to get the low observed  \nmetallicity in,  for example, old clusters and \ninter-stellar clouds. This could be in the form of a generation \nof very short-lived type III stars. Such a generation of stars\nmay also be necessary to ionize the intergalactic medium. \nThe extrapolation of $^4He$ \nabundance in type II objects and low metal (HII) galaxies, \nto its zero heavy metal abundance \nlimit, presupposes that reprocessing and production of heavy\nelements in type III stars is not accompanied by a significant \nchange in the $^4He$ levels. A violation of this assumption,\ni.e. a minute increase in $^4He$ during reprocessing (even as low\nas 1 - 2 \\%) would rule out the minimal SBBN. As a matter of fact, \nit is possible to account for the entire pre-galactic $^4He$ \nby such objects \\cite{wagoner}. \n\n\nFinally, of late \\cite{lem}, the need for a careful scrutiny and \na possible revision of the status of SBBN has also been suggested \nfrom  the reported high abundance of $D$ in several $Ly_\\alpha$ \nsystems. It may be difficult to accommodate such high abundances\nwithin the minimal SBBN. Though the status of these observations \nis still a matter of debate, and (assuming their confirmation) \nattempts to reconcile the cosmological abundance of deuterium \nand the number of \nneutrino generations within the framework of SBB are still on, \na reconsideration of alternate routes to deuterium in a slow\nexpanding universe as described in this article\ncould well be worth the effort. This is specially in consideration\nof the stranglehold that Deuterium has on SBB in constraining the \nbaryon density upper limit to not more than some 3 to 4 \\% . This\nconstraint has been used in SBB to make out a strong case for non -\nbaryonic dark matter to make up the mass estimates at galactic and\ncluster scales. Relying on Deuterium that is so local environment\nsensitive, to predict the nature of CDM runs the risk of ``building\na colossus on a few feet of clay''\\cite{borner}.\n\nThis article reports our study of nucleosynthesis in a universe \nin which the scale factor evolves linearly with time independent \nof the equation of state of matter. A strictly linear evolution \nof the cosmological scale factor\nis surprisingly an excellent fit to a host of cosmological \nobservations. Any model that can support such a coasting presents \nitself as a falsifiable model as far as classical cosmological \ntests are concerned as it exhibits distinguishable and verifiable \nfeatures. \n\nLarge scale homogeneity and isotropy observed in the universe is\nincorporated in the Friedman-Robertson-Walker (FRW) metric:\n\\begin{equation}\n\\label{1}\nds^2 = dt^2 - a(t)^2[{dr^2\\over {1 - Kr^2}} + r^2(d\\theta^2 \n+ sin^2\\theta d\\phi^2)]\n\\end{equation}\nHere $K = \\pm 1, 0$ is the curvature constant. In the following \nsection, we summarize the \nconcordance of observations in a $K = -1$ FRW cosmology in which \nthe scale factor evolves linearly \nwith time: $a(t) \\propto t$, right from the creation event itself. \nThe motivation for such an endeavor has been discussed at \nlength in a series of earlier articles \\cite{meetu,annu}. \nFor the purpose of this article, we shall take as a conjecture\nthat a strictly homogeneous background FRW model coasts \nlinearly with time. This can be achieved in a large class of \nnon-minimally coupled theories of gravity. However, perturbations\naround the homogeneous background are asumed to satisfy the \nperturbed Einstein equations. \nSection 3 describes concordant nucleosynthesis in such a model.\n\n\\section{Concordance of a linear coasting cosmology:}\n\n{\\bf \\noindent Classical Cosmology tests}\n\\begin{itemize}\n\\item{\\it\\bf \\noindent n(z), a(z):} Data on Galaxy number counts \nas a function of red-shift  along with data angular diameter \ndistance as a function of red-shift do not rule out a linearly \ncoasting cosmology \\cite{kolb1}. However, as these tests are \nmarred by evolutionary effects (and mergers), they have fallen \ninto disfavour as reliable tests of a viable model.\n\n\n\\item{\\it\\bf \\noindent Hubble Diagram:} With the discovery of \nSupernovae type Ia [SNe Ia] as reliable standard candles, the \nstatus of the Hubble test has been elevated to that of a precision\nmeasurement. The Hubble plot relates the magnitude of a standard \ncandle to its redshift in an expanding FRW universe. In \n\\cite{meetu} we demonstrated how linear coasting is as \naccommodating for high red-shift objects as the standard \nnon--minimal model with a small cosmological constant. \nThe concordance of linear coasting with SNe1a data\nfinds a passing mention in the analysis of Perlmutter \\cite{perl}\nwho noted that the Hubble plot for $\\Omega_\\Lambda = \\Omega_M = 0$\n(for which the scale factor would have a linear evolution in \nstandard cosmology) is ``practically identical to the \n$\\bf{best fit}$ plot for an unconstrained cosmology''. The \nconcordance of this Hubble plot continues even for \nthe  more recent data on SNe1a with redshifts $z \\ge 1$. \nThe plot almost coincides with the Hubble plot for  \n$\\Omega_\\Lambda \\approx 0.72,~ \\Omega_M = 0.28$.\n\n\\item{\\it\\bf \\noindent Age of the Universe} The age estimate of \nan ($a(t) \\propto t$) universe, deduced from a measurement of the \nHubble parameter, is given by $t_o = (H_o)^{-1}$.  The low \nred-shift SNe1a give the best value of \n$65~{\\rm km~ sec^{-1}~ Mpc^{-1}}$ for the Hubble parameter.\nThe age of the universe turns out to be $15\\times 10^9$ years.\nSuch an age estimate is comfortably concordant with age estimates \nof old clusters. \n\n\\item{\\it\\bf \\noindent Lensing Statistics:} Consistency and\nconcordance of linear \ncoasting with gravitational lensing statistics was reported in \n\\cite{abha}. \n\n\\item{\\it \\bf \\noindent  Density Perturbations }\n\\vspace{.2cm}\n\nIn \\cite{pranav} we explored a conjecture that the background\nuniverse coasts linearly with perturbations around this \nbackground being described by\n\\begin{equation}\n-8\\pi G \\delta T^M_{\\mu\\nu} = \\delta G_{\\mu\\nu}\n\\end{equation}\nIt turns out that small perturbations can \nevolve to a non-linear regime and therefore be expected to lead \nto structures at large scales. This can be seen by \nexpressing the metric as\n$ g_{\\mu\\nu} = ^{(o)} g_{\\mu\\nu} + \\delta g_{\\mu\\nu}$, \nthe $\\delta g_{\\mu\\nu}$ being the perturbations. Scalar\nperturbations can be decomposed in terms of eigenmodes of\nthe laplacian on the constant $\\bar{\\eta}$ surface \n(here $d\\bar{\\eta} = a^{-1}dt$) with eigenvalues $-k^2$\nIn terms of matter gauge invariant variable  $D_g$\n(see eg. \\cite{mukh,ruth}) and a density parameter\n$$\nC \\equiv 4\\pi G \\rho_b^o a_0^2 = {3\\over 2}{{8\\pi G}\\over {3H_o^2}}\\rho_b^o \n= {3\\over 2}\\Omega_b  \n$$\nthe density perturbation equation simply reduces to \\cite{pranav}:\n\\begin{equation}\n[(k^2 + 3){{e^{\\bar{\\eta}}}\\over C} + 3]\\ddot D_g +\n[(k^2 + 3){{e^{\\bar{\\eta}}}\\over C} + 2]\\dot D_g\n- D_gk^2  = 0.\n\\end{equation}\n$k = 1$ corresponds to the Hubble scale which is the same as \nthe curvature scale in this model. At a redshift $\\approx 1000$, \na sphere of Hubble radius subtends an\nangle roughly .25 degrees. Using constraints from microwave \nbackground anisotropy at these angles gives $D_g \\approx 10^{-5}$ \nat these scales at the last scattering surface. It is easy to see \nthat modes \n$k \\le 1$ do not grow. At smaller angular scales ($k >> 1$),  \nthe observed anisotropy is expected to fall to much lower values \n\\cite{savthes}. Photon diffusion dampens anisotropies at angular \nscales smaller than about one minute. However, for such large \nvalues of $k$, $D_g$ has rapidly growing solutions. The \nperturbation equation becomes  \n\\begin{equation}\n\\ddot D_g + \\dot D_g - Ce^{-\\bar{\\eta}}D_g = 0  \n\\end{equation}\nThis has exact solutions in terms of modified first and second \ntype bessel functions $I_1,~K_1$:\n\\begin{equation}\nD_g = C_1(Ce^{-\\bar{\\eta}})^{1\\over 2}I_1((4Ce^{-\\bar{\\eta}})^{1\\over 2})\n       + C_2(Ce^{-\\bar{\\eta}})^{1\\over 2}K_1((4Ce^{-\\bar{\\eta}})^{1\\over 2})\n\\end{equation}\nFor large arguments, these functions have their asymptotic forms:\n\\begin{equation}\nI_1 \\longrightarrow {(Ce^{-\\bar{\\eta}})^{-{1\\over 4}}\\over {2\\sqrt{\\pi}}}\nexp[2(Ce^{-\\bar{\\eta}})^{1\\over 2}];~~\nK_1 \\longrightarrow {(Ce^{-\\bar{\\eta}})^{-{1\\over 4}}\\over {2\\sqrt{\\pi}}}\nexp[- 2(Ce^{-\\bar{\\eta}})^{1\\over 2}]\n\\end{equation}\nEven if diffusion damping were to reduce the baryon density contrast to \nvalues as low as some $10^{-15}$, a straight forward numerical integration\nof eqn(4) demonstrates that for $k \\ge 3000$ the density contrast becomes \nnon linear around redshift of the order 50. \n\nIn contrast to the above, in the radiation dominated epoch,\nthe adiabatic approximation perturbation equations imply \n\\cite{pranav}:\n\\begin{equation}\n[(k^2 + 3){3\\over {4k^2}} + {3\\tilde C\\over {2k^2e^{2\\bar{\\eta}}}}]\\ddot D_g +\n{3\\tilde C\\over {k^2e^{2\\bar{\\eta}}}}\\dot D_g\n+[{{k^2 + 3}\\over 8} - {\\tilde C\\over {2e^{2\\bar{\\eta}}}}]D_g  = 0 \n\\end{equation}\nFor $\\bar{\\eta}$ large and negative, small $k$ pertubation equation \nreduces to \n\\begin{equation}\n3\\ddot D_g + 6\\dot D_g -k^2D_g = 0\n\\end{equation}\nEqns(7-8) imply that \nperturbations bounded for large negative $\\eta$ damp out for \nsmall $k$ while large $k$ modes are oscillatory.\n\nWe conclude that fluctuations do not grow in the radiation \ndominated era, small $k$ (large scale) fluctuations do not grow \nin the matter dominated era as well. However, \neven tiny residual baryonic fluctuations $O(10^{-15})$ \nat the last scattering surface for large values of $k \\ge 3000$ \nin the matter dominated era, grow to the non linear regime. \nSuch a growth would be a necessary condition for structure \nformation and is not satisfied in the standard model. In the \nstandard model, cold dark matter is absolutely essential \nfor structure formation. \n\n\n\\vspace{.2cm}\n\\item{\\it \\bf \\noindent The recombination epoch}\n\\vspace{.2cm}\n\n        Salient features of the plasma era in a linear coasting \ncosmology have been described in \\cite{astroph,savitaI,savthes}. \nHere we reproduce some of\nthe peculiarities of the recombination epoch. These\nare deduced by making a simplifying assumption of thermodynamic \nequilibrium just before recombination. \n\nThe probability that a photon was last scattered in the interval \n$(z,z + dz)$ can be expressed in terms of optical depth,and turns out to be:\n\\begin{equation}\nP(z) = e^{-\\tau_\\gamma}{{d\\tau_\\gamma}\\over {dz}} \\approx 7.85\\times 10^{-3}\n({z\\over {1000}})^{13.25}exp[-0.55({z\\over {1000}})^{14.25}]\n\\end{equation}\nThis $P(z)$ is sharply peaked and well fitted by a gaussian of mean redshift\n$z \\approx 1037$ and standard deviation in redshift $\\Delta z \\approx 67.88$.\nThus in a linearly coasting cosmology, the last scattering surface locates at\nredshift $z^* = 1037$ with thickness \n$\\Delta z \\approx 68$. Corresponding values in \nstandard cosmology are $ z = 1065$ and $\\Delta z \\approx 80$.\n\n   An important scale that determines the nature of CMBR anisotropy is the \nHubble scale which is the same as the curvature scale for linear coasting. \nThe angle subtended today, by a sphere of Hubble radius at $z^* = 1037$, \nturns out to be $\\theta_H \\approx 15.5$ minutes. The Hubble length determines the scale \nover which physical processes can occur coherently. Thus one expects all \nacoustic signals to be contained within an angle $\n\\theta_H \\approx 15.5$ minutes.\n\nWe expect the nature of CMB anisotropy to follow from the above \nresults. The details are still under study and shall be reported \nseparately.\n\n\\vspace{.5cm}\n\\item{\\it \\bf \\noindent Summary}\n\\vspace{.5cm}\n\nIn spite of a significantly different evolution, a linear \ncoasting cosmology can not be ruled out by all the tests we have \nsubjected it to so far. Linear coasting being extremely\nfalsifiable, it is encouraging to observe its\nconcordance !! In standard cosmology, falsifiability has taken \na backstage - one just constrains the values of cosmological \nparameters subjecting the data to Bayesian statistics. Ideally, \none would have been very content with a cosmology based on physics\ntested in the laboratory. Clearly, standard cosmology does not \npass such a test. One needs a mixture of hot and cold dark \nmatter, together with (now) some form of \\emph{dark energy} to \nact as a cosmological constant, to \nfind any concordance with observations. In other words, one uses \nobservations to parametrize theory in Standard Cosmology. \nIn contrast, a universe that is born and evolves as a curvature \ndominated model has a tremendous concordance, it does not need any form of dark matter and there are \nsufficient grounds to explore models that support such a coasting.\n\n\\end{itemize}\n\\section{The Nucleosynthesis Constraint:}\n\nWhat makes linear coasting particularly appealing is a \nstraightforward adaptation of standard nucleosynthesis codes to\ndemonstrate that primordial nucleosynthesis is not an impediment \nfor a linear coasting cosmology \\cite{kapl,annu}. A linear \nevolution of the scale factor radically effects nucleosynthesis\nin the early universe. With the present age of the universe some \n$15\\times 10^9$ years and the $effective$ CMB temperature 2.73 K, \nthe universe turns out to be some 45 years old at $10^9$ K.  \nWith the universe expanding at such low rates, weak interactions \nremain in equillibrium for temperature as low as $\\approx 10^8$ K.\nThe neutron to proton ratio is determined by the n-p\nmass difference and is approximately $n\/p\\sim exp[-15\/T_9]$.\nThis falls to abysmally low values at temperatures below $10^9$ K.\nSignificant nucleosynthesis leading to helium formation commences \nonly near temperatures below $\\sim 5\\times 10^9$K. The low n\/p\nratio is not an impediment to adequate helium production. This\nis because once nucleosynthesis commences, inverse beta decay \nreplenishes neutrons by converting protons into neutrons and \npumping them into the nucleosynthesis network. For baryon entropy \nratio $\\eta\\approx 7.8\\times 10^{-9}$, the standard \nnucleosynthesis network can be modified for linear coasting and gives $\\approx 23.9\\% $ Helium.\nThe temperatures are high enough to cause helium to burn.\nEven in SBBN the temperatures are high enough for helium to burn.\nHowever, the universe expands very rapidly in SBBN. In comparison,\nthe linear evolution gives enough time for successive burning \nof helium, carbon and oxygen. The metallicity yield is some $10^8$ times the metallicity \nproduced in the early universe in the SBBN. The metallicity \nis expected to get distributed amongst nucleii with maximum \nbinding energies per nucleon. These are nuclei with atomic \nmasses between 50 and 60. This metallicity is close to\nthat seen in lowest metallicity objects. Figure(1) \\& (2) describe \nnuclesynthesis as a function of the Baryon entropy ratio. The \nmetallicity concommitantly produced with $\\approx 23.9\\%$ \nHelium is roughly $10^{-5}$ solar. \n\nThe only problem that one has to contend with is the significantly\nlow residual deuterium in such an evolution. The desired amount\nwould have to be produced by the spallation processes much later \nin the history of the universe as described below.\n\nInterestingly, the baryon entropy ratio required for the right \namount of helium corresponds to $\\Omega_b \\equiv \\rho_b\/\\rho_c = \n8\\pi G \\rho_b\/3H_o^2 \\approx 0.69$ . This closes dynamic mass \nestimates of large galaxies and clusters [see eg \n\\cite{tully}]. In standard cosmology this closure is \nsought by taking recourse to non-baryonic cold dark matter. \nThere is hardly any budget for non - baryonic CDM in linear \ncoasting cosmology.\n\n\n\n{\\bf Deuterium Production:}\n\nTo get the observed abundances of light elements besides $^4He$,\nwe recall spallation mechanisms that were explored in the \npre - 1976 days \\cite{eps}. Deuterium can indeed be produced by \nthe following spallation reactions:\n$$\np + ^4He \\longrightarrow D + ^3He; ~~ 2p \\longrightarrow D + \\pi^+;\n$$\n$$\n2p \\longrightarrow 2p + \\pi^o,~ \\pi^o \\longrightarrow 2\\gamma,~\n\\gamma +^4He \\longrightarrow 2D.\n$$There is no problem in producing Deuterium all the way \nto observed levels. The trouble is that under most conditions \nthere is a concomitant over - production of $Li$ nuclei and \n$\\gamma$ rays at unacceptable levels. Any later destruction of \nlithium in turn completely destroys $D$. As described in \n\\cite{eps}, figure (3) exhibits relative production of $^7Li$  \nand $D$ by spallation. It is apparent that the production of \nthese nuclei to observed levels and without a collateral \ngamma ray flux is possible only if the incident (cosmic ray or any\nother) beam is energized to an almost mono energetic value of \naround 400 MeV. A model that requires nearly mono energetic \nparticles would be rightly considered \n$ad~hoc$ and would be hard to physically justify.\n\nHowever, lithium production occurs by spallation of protons over \nheavy nuclei as well as spallation of helium over helium:\n$$\np,\\alpha ~+ ~C,N,O \\longrightarrow Li~+~X;~~ \np,\\alpha ~+ ~Mg,Si,Fe \\longrightarrow Li~+~X;~~\n$$\n$$\n2\\alpha \\longrightarrow ^7Li ~+~p; ~~ \\alpha ~+~D \\longrightarrow p ~+~^6Li;\n$$\n$$\n^7Be + \\gamma \\longrightarrow p + ^6Li; ~~ ^9Be + p \\longrightarrow\n\\alpha +^6Li.\n$$\nThe absence or deficiency of heavy nuclei in a target cloud and \ndeficiency of alpha particles in the incident beam would clearly \nsuppress lithium production. Such conditions could well have \nexisted in the environments of incipient Pop II stars. \n\nEssential aspects of evolution of a collapsing cloud to form a low\nmass Pop II star is believed to be fairly well understood \n\\cite{feig,hart}. The formation\nand early evolution of such stars can be discussed in terms of\ngravitational and hydrodynamical processes. A protostar would \nemerge from the collapse of a molecular cloud core and would be \nsurrounded by high angular momentum material forming a \ncircumstellar accretion disk with bipolar outflows.\nSuch a star contracts slowly while the magnetic fields play a \nvery important role in regulating collapse of the accretion disk \nand transferring the disk orbital angular motion to collimated \noutflows. A substantial fraction of the accreting matter is \nejected out to contribute to the inter - stellar medium.\n\nEmpirical studies of star forming regions over the last twenty \nyears have now provided direct and ample evidence for MeV \nparticles produced within protostellar and T Tauri systems \n\\cite{Terekhov,Torsti}. The source of such accelerated \nparticle beaming is understood to be violent magnetohydrodynamic \n(MHD) reconnection events. These are analogous to solar magnetic\nflaring but elevated by factors of $10^1$ to $10^6$ above levels \nseen on the contemporary sun besides being up to some 100 times \nmore frequent. Accounting for characteristics in the meteoritic \nrecord of solar nebula from integrated effects of particle \nirradiation of the incipient sun's flaring has assumed the status \nof an industry. Protons are the primary component of particles \nbeaming out from the sun in gradual flares while $^4He$ are \nsuppressed by factors of ten in rapid flares to factors of a \nhundred in gradual flares\\cite{Terekhov,Torsti}. Models of young \nsun visualizes it as a much larger protostar with a cooler \nsurface temperature and with a very highly elevated level of \nmagnetic activity in comparison to the contemporary sun. It is \nreasonable to suppose that magnetic reconnection events would \nlead to abundant release of MeV nuclei and strong shocks that \npropagate into the circumstellar matter. Considerable evidence \nfor such processes in the early solar nebula has been found in \nthe meteoric record. It would be fair to say that the \nhydrodynamical paradigms for understanding the earliest stages of \nstellar evolution is still not complete. However, it seems \nreasonable to conjecture that several features of collapse of a \ncentral core and its subsequent growth from acreting material \nwould hold for low metallicity Pop II stars. Strong magnetic \nfields may well provide for a link between a central star, its \ncircumstellar envelope and the acreting disk. Acceleration of \njets of charged particles from the surface of such stars could \nwell have suppressed levels of $^4He$. Such a suppression \ncould be naturally expected if the particles are picked up from \nan environment cool enough to suppress ionized $^4He$ in \ncomparison to ionized hydrogen. Ionized helium to hydrogen number \nratio in a cool sunspot temperature of $\\approx 3000~K$ can be \ncalculated  by the Saha's ionization formula and the \nionization energies of helium and hydrogen. This turns out to be \n$\\approx~ exp(-40)$ and increases rapidly with temperature. Any \nelectrodynamic process that accelerates charged particles from \nsuch a cool environment would yield a beam deficient in alpha \nparticles. With $^4He$ content in an accelerated particle beam \nsuppressed in the incident beam and with the incipient cloud \nforming a Pop II star having low metallicity in the \ntarget, the ``no - go'' concern of (Epstein et.al. \\cite{eps}) \nis effectively circumvented. The ``no-go'' used \n$Y_\\alpha \/Y_p \\approx .07$ in both the energetic particle\nflux as well as the ambient medium besides the canonical solar \nheavy element mass fraction. Incipient Pop II environments may \ntypically have heavy element fraction suppressed by more than \nfive orders of magnitude while, as described above, magnetic \nfield acceleration could accelerate beams of particles deficient \nin $^4He$.\n\nOne can thus have a broad energy band - all the way from a few \nMeV up to some 500 MeV per nucleon as described in the Figure (3), \nin which acceptable levels of deuterium could be ``naturally'' \nproduced. The higher energy end of the band may also\nnot be an impediment. There are several astrophysical processes \nassociated with gamma ray bursts that could produce $D$ at high \nbeam energies with the surplus gamma ray flux a natural by product.\n\n\n\nCircumventing the ``no-go'' concern of Epstein et al would be\nof interest for any cosmology having an early universe \nexpansion rate significantly lower than corresponding rates\nfor the same temperatures in early universe SBB.\n\n{\\bf Conclusions:}\n\nOur understanding of star formation has considerably evolved \nsince 1976. SBBN constraints need to be reconsidered in view of \nempirical evidence from young star forming regions. These models \nclearly imply that spallation mechanism can lead to viable and \nnatural production of Deuterium and Lithium in the incipient \nenvironment of Pop II stars. One can conceive of cosmological \nmodels in which early universe nucleosynthesis produces the \ndesired primordial levels of $^4He$ but virtually no $D$. Such a \nsituation can arise in SBBN itself with a high baryon entropy \nratio $\\eta$. In such a universe, in principle,\nDeuterium and Lithium  can be synthesized up to acceptable levels \nin the environment of incipient Pop II stars.\n\n\nIn SBB, hardly any metallicity is produced in the very early \nuniverse. Metal enrichment is supposed to be facilitated by a \ngeneration of Pop III stars. Pop III star formation from a \npristine material is not well understood till date in spite of a \nlot of effort that has been expanded to that effect recently \n\\cite{sneider}. It is believed that with metallicity below a \ncritical transition metallicity \n($Z_{cr} \\approx 10^{-4} Z_\\odot$), masses of Pop III stars would \nbe biased towards very high masses. Metal content higher than \n$Z_{cr}$ facilitates cooling and a formation of lower mass Pop II \nstars. In SBB, the route to Deuterium by spallation discussed in \nthis article would have to follow a low metal contamination by a \ngeneration of Pop III stars.\n\n\nDeuterium production by spallation discussed in this article \nwould be good news for a host of slowly evolving cosmological \nmodels \\cite{kapl,annu}. An FRW model with a linearly evolving \nscale factor enjoys concordance with constraints on age of the \nuniverse and with the Hubble data on SNe1A. Such a linear coasting\nis consistent with the right amount of helium\nobserved in the universe and metallicity yields close to the \nlowest observed metallicities. The only problem that one has to \ncontend with is the significantly low yields of deuterium in such \na cosmology. In such a model, the first generation of stars would \nbe the low mass Pop II stars and the above analysis would \nfacilitate the desired deuterium yields.\n\n\nIn SBB, large-scale production and recycling of metals through \nexploding early generation Pop III stars leads to verifiable \nobservational constraints. Such stars would be visible as \n27 - 29 magnitude stars appearing any time in every square \narc-minute of the sky. Serious doubts have been expressed on the \nexistence and detection of such signals \\cite{escude}. The linear \ncoasting cosmology would do away with the requirement of such \nPop III stars altogether.\n\n\\begin{center}\n\\begin{figure}\n\\resizebox{.8\\columnwidth}{!}\n{\\includegraphics[angle=270]{fig1a.ps}}\\\\\n\\title{Fig1(a). The figure shows abundance of He4 as a function of temperature for $\\eta\\approx 7.8\\times 10^{-9}$. The final abundance of He4 is approximately 23 \\%. It reaches this value around $T \\approx T_9$ and stays same thereafter.}\\\\\n\\resizebox{.8\\columnwidth}{!}\n{\\includegraphics[angle=270]{fig1b.ps}}\\\\\n\\title{Fig1(b). The figure shows metalicity as a function of temperature for $\\eta\\approx 7.8\\times 10^{-9}$. The metallicity for a linaer coasting model is nearly equal to $ 10^{-5}$  times solar metallicity. }\\\\ \n\\end{figure}\n\\end{center}\n\\begin{center}\n\\begin{figure}\n\\resizebox{.8\\columnwidth}{!}\n{\\includegraphics[angle=270]{fig2a.ps}}\\\\\n\\title{Fig2(a). The figure shows He4 abundance as a function of $\\eta$.}\\\\\n\\resizebox{.8\\columnwidth}{!}\n{\\includegraphics[angle=270]{fig2b.ps}}\\\\\n\\title{Fig2(b). The figure shows metallicity as a function of $\\eta$.}\\\\\n\\end{figure}\n\\end{center}\n\\eject\n\\begin{center}\n\\begin{figure}\n\\resizebox{.8\\columnwidth}{!}\n{\\includegraphics{fig3.ps}}\\\\\n\\title{ Fig3. The rates at which abundances approach their present values as a function of the energy per nucleon of the incident particle.} \\cite{eps}\\\\\n\\end{figure}\n\\end{center}\n\\vskip 1cm\n\n\n\n\\vfil\\eject\n{\\bf Acknowledgment.} \\\\ Daksh Lohiya and Sanjay Pandey acknowledge IUCAA for support under \nthe IUCAA Associate Program. Geetanjali and Pranav ackowledge C.S.I.R for financial support. \\\\\n\\vspace{0.5cm}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}