On Sign Change Of Fourier Coefficients Of Cusp Forms

Dirichlet's theorem says that there are infinitely many primes in the arithmetic progression n = 1(mod q)with(q,l)= 1.It is a natural question that how big the least prime in this arithmetic progression,denote by P(q,l).Linnik[29,30]proved that there is an absolute constant L>0 such that P(q,l)(?)ql,this constant L was called Linnik's constant and this question was called Linnik's problem.After that,many authors established acceptable numerical values for Lin-nik's constant L.There are many problems that can be formulated similar as Linnik's problem,which are called Linnik-type problems in number theory.We study some of Linnik-type problems in this paper.Suppose that q ? 2 be an integer,X be a non-principle Dirichlet character modulo q.Let nx be the least positive integer for which X(n)?0,1,we refer to the evaluation of nx as Linnik-type problem.In case x coincides with the Legen-dre symbol,nx is a least quadratic non-residue.The study of the least quadratic non-residue has an almost century-long history,see[4,37,39,56].We can con-sider characters as GL(1)objects,one may wonder what the GL(2)(or even GL(n))analogues are for this problem.Here,we have three questions toward this problem:· the first negative coefficients of automorphic L-functions for GL(n)with n ?2:? how many sign changes of coefficients of automorphic L-functions for GL(n)with n ? 2 in a long interval[1,x];·the behavior on the signs of coefficients of automorphic L-functions for GL(n)with n>2 in short intervals[x,x ? xr],or even more for some special se-quences.There are many authors who contributed to these problems,see[23,24,34,38,40,41,51,52,57,58].In this paper,we consider the following three questions and get some new results.Firstly,we consider the first sign change of Fourier coefficients of general cusp forms,see Theorem 0.1,this question was first studied by Choie and Kohnen[6].Secondly,we consider the signs of Fourier coefficients of primitive cusp forms for GL(2)in short interval over sparse sequence,see Theorem 0.2.At last,We consider simultaneous sign changes of Fourier coefficients of two different primitive cusp forms for GL(2)in short interval,see Theorem 0.3.Let Sk(N)(resp.Snewk(N))be the space of cusp form(resp.primitive cusp form)of even integral weight k? 2 on the Hecke congruence subgroup ?o(N)(?)S L2(Z).Denote aF(n)(resp.?f(n))be the n-th normalized Fourier coefficient of F ? Sk(N)(resp.f ? Snewk(N)).Suppose that aF(n)and ?f(n)are both real.Then we have the first main theorem.Theorem 0.1 Let F ? Sk(N)be a nonzero cusp form with N square-free.Let aF(n)be the n-th normalized Fourier coefficient of F.For any ?>0,there exist n1,/n2 with n1.n2(?)(kN)2+?,(0.2)such that aF(ni)aF(n2)<0.Next,we consider the sign changes of Fourier coefficients of f ? Snewk(N)in short interval over spare sequence and get the following theorem.Theorem 0.2 Let f ?Snewk(N)be a nonzero primitive cusp form with N square--ree.Let ?f(n)be the n-th normalized Fourier coefficient of.Then for any r with 13/17<r<1,at least one sign change forr ?f(n2)(n? 1)occurs with n(x,x+xr]or sufficiently large x.Moreoverr the number of sign changes for n?x is(?)x-r.We can also consider simultaneous sign changes of Fourier coeefficients of two ifferent primitive cusp forms in short interval and et the following theorem.Theorem 0.3 Let f,g E Snewk(N)be two different nonzero cusp forms with N square-free.Let ?f(n)(resp.?g(n))be the n-th normalized Fourier coefficient of f(resp.g).Then for any r with 13/15<r<1,at least one sign change for?f(n)?g(n)(n ? 1)occurs with n ?(x,x? xr]for sufficiently large x.Moreover,the number of sign changes for n ? x is(?)x1-r.