Some Research On The Mean Value Of Fourier Coefficients Of Cusp Forms

Many well-known problems in number theory can be reduced to study the asymptotic formula of the sum of an arithmetic function a(n),namely,S(x)=(?)a(n).That is to find the main term of the sum function and estimate the order of the error term as well as possible.For example,the distribution of prime numbers in natural numbers is reduced to the sum of ?(n),Dirichlet divisor problem is reduced to the sum of d(n),Gauss circle problem is reduced to the sum of r(n),the distribution of square-free numbers in natural numbers can be reduced to the discussion of the sum of ?2(n).In the analytic number theory,there are several common types to estimate the sum of arithmetic function a(n).For instance,(1)Dirichlet convolution sums(?)a1(n1)a2(n2)…ak(nk).(2)Shifted convolution sums(?)a1(n+h1)a2(n+h2)…ak(n+hk),where h1,h2,…,hk are integers.(3)High order mean estimation on the polynomial with integer coefficients(?)ak(|f(n)|),where k?1 is an integer,f(n)is a polynomial with integer coefficients.The common divisor problem can be viewed as the type of Dirichlet convolu-tion sums.Let ?(s)be the Riemann zeta-function,when Re s>1,one has?k(s)=(?)dk(n)/ns.Landau,Voronoi,Hardy,Littlewood studied(?)1:(?)dk(n)=xpk(log x)+O(x?k+?),where Pk(x)is a polynomial of degree k-1,?k<1.The well-known Chowla conjecture can be regarded as the type of shifted con-volution sums.Let A(n)be the Liouville function,for any distinct natural numbers h1,…,hk,k>1,one has(?)?(n+h1)…?(n+hk)=o(X),where X??Matomaki-Radziwill-Tao[30]have established an averaged version of this conjecture,namely(?)where H?H(X)? X,k is fixed,when X??,H(X)??.There are also many studies about high order mean estimation on the poly-nomial with integer coefficients in the number theory.Let f(z)be the normalized primitive holomorphic form for the full modular group SL(2,Z),and f(z)has a Fourier expansion at the cusp infinity.Let ?f(n)be the Fourier coefficients of f(z),Hecke,Selberg,Rankin,Moreno,Shahidi,Fomenko,Lau,Wu,Lii studied the asymptotic behavior of the high power of ?f(n),(?)?fk(n)=xPk(log x)+O(x?k),where k ?8,?k<1,when k is odd,Pk(x)? 0;when k is even,P2(x),P4(x),P6(x),P8(x)are polynomials of degree 0,1,4,13.It is an important topic to estimate sums of Fourier coefficients of automorphic cusp forms,which has many applications in number theory.Such as,the oscil-lation and distribution of Fourier coefficients of cusp forms,subconvexity bound and unique ergodicity.In this paper,we focus our attention to three types of mean value of Fourier coefficients of Hecke-Maass cusp forms and holomorphic cusp forms for the full modular group SL(2,Z).The main contents of this paper are as follows:Dirichlet convolution sums of Fourier coefficients of Hecke-Maass cusp forms,averages of shifted convolution sums of Fourier coefficients of holomorphic cusp forms,and application of the quadratic mean over arithmetic progressions of Fourier coefficients of holomorphic cusp forms.First of all,for the Dirichlet convolution sums of Fourier coefficients of Hecke-Maass cusp forms,let f(z)be the normalized primitive Hecke-Maass forms for the full modular group SL(2,Z),and let ?f(n)be the Fourier coefficients of f(z).We concentrate on the upper bound of Dirichlet convolution sums of Fourier coeffi-cients ?f(n),namely,Sk(x):=(?)?f(n1)…?f(nk),k?2.For the case of k=1,Jiang-Lu[33]have given the best result at present,(?)Let f(z)be the normalized primitive Hecke-Maass forms for the full modular group SL(2,Z),and let ?f×f(n)be the coefficients of the Rankin-Selberg L-function L(s,f×f)attached to f(z).We concentrate on improving Dirichlet convolution sums of coefficients-?f×f(n),namely,(?)?f×f(n1)…?f×f(nk),k?2.For the case of k=1,by the method of Rankin-Selberg,we have the best result at present,(?)?f×f(n)=Cfx+Of(x3/5),where Cf is a certain constant.Our research idea stems from Jiang-Lu[34],which established a general sum-mation formula for the coefficients of a class of L-functions,without assuming the generalized Ramanujan conjecture.As an application,we study Dirichlet convo-lution sums of Fourier coefficients of Hecke Maass cusp forms.Our results are as follows.Theorem 0.1 Let f(z)be the normalized primitive Hecke-Maass forms for the full modular group SL(2,Z),and f(z)has a Fourier expansion at the cusp infinity.Let?f(n)be the Fourier coefficients of f(z),and let ?f×f(n)be the coefficients of the Rankin-Selberg L-function L(s,f×f)attached to f(z).Define Sk(x):=(?)?f(n1)…?f(nk),k?2.For any ?>0,we have S2(x)<<f,? x1/2 log 2 x,S3(X)<<f,?x 148/243+?,for 4 ?k?20,Sk(x)<<f,? x5/8+79k-316/216k-108+?,for k>20,Sk(x)<<f,? x1-3/2k+?,where the implied constant depends on f and ?.Define Uk(x):=(?) ?f×f(n1)…?f×f(nk),k?2.For any ?>0,we have(?)where Nk(x)has the form of xPk(log x)and Pk(x)is a polynomial of degree k-1,and the O-constant depends on f and ?.Secondly,for shifted convolution sums of Fourier coefficients of holomorphic cusp forms,we mainly focus on our attention to averages of shifted convolution sums of Dirichlet convolution of Fourier coefficients ?f(n)of holomorphic cusp forms for the full modular group SL(2,Z),namely,(?)?k,f(n)?k,f(n+h),Ak,f(n)=(?)?f(n1)…?f(nk).Our research idea comes from Baier-Browning-Marasingha-Zhao[5],which is about averages of shifted convolutions of d3(n).Our results are as follows.Theorem 0.2 Let f(z)be the normalized primitive holomorphic forms for the full modular group SL(2,Z),and f(z)has a Fourier expansion at the cusp infinity.Let Af(n)be the Fourier coefficients of f(z).Define?k,f(n)=(?)?f(n1)…?f(nk),For any ?>0,1 ?H?N,we have(?)?2f(n)?2,f(n+h)<<N6/5+?H2/5,N1/3?H?1-?,(?)?3,f(n)?3,f(n+h)<<N4/3+?H1/3,N1/2 ? H?N1-?,for k?4,(?) ?k,f(n)?k,f(n+h)<<N4k+1/2k+4+?H2/k+2,N2k-3/2k?H?N1-?,where the implied constant depends on f and ?.At last,we study the application of quadratic mean over arithmetic progres-sions of Fourier coefficients of holomorphic cusp forms.We first study the quadrat-ic mean of normalized Fourier coefficients of holomorphic cusp forms over arith-mecic progressions using the order estimation of L(s,?),L(s,sym2f(?)?),and ?(s).Then by the method of Erdos-Ivic[15],which studied the functions with square kernel,we study shifted convolution sums of normalized Fourier coefficients of holomorphic cusp forms and the functions with square kernel.Our results are as follows.Theorem 0.3 Let f(z)be the normalized primitive holomorphic forms for the full modular group SL(2,Z),and f(z)has a Fourier expansion at the cusp infinity.Let?f(n)be the Fourier coefficients of f(z).Every integer n?1 may be uniquely decomposed as n=q(n)s(n),(q(n),s(n))=1,where q(n)is square-free and s(n)is square-full(s(n)is square-full if p2|s(n),whenever p|s(n)).Let a(n)be the function with square-full kernel satisfying a(n)=a(s(n)),a(n)<<n?.For any ?>0,we have(?)a(n)?f2(n+1)=Cx+O(x13/14+?),where C is a constant which may be explicitly evaluated,and the O-constant de-pends on f and ?.