The Distribution Of Fourier Coefficients Of Cusp Form Over Arithmetic Progressions

Fourier coefficients of holomorphic cusp form and Maass form are of great interest to many scholars.In this paper,we combine the classical analytic number theory method with automorphic L-function theory to study the distribution of the Fourier coefficients of the Holomorphic cusp form and the Maass form in the arithmetical sequence on the full modular group ?=SL2(Z).Let Hk denote the set of normalized primitive holomorphic cusp forms of even integral weight k.The Fourier expansion of f?Hk Hk at the cusp ? is(?)where the coefficients ?f(n)? R are eigenvalues of Tn.For all n? 1,Deligne's[8]bound is|?f(n)|?d(n)where d(n)is the divisor function.In 1990,Rankin[33]obtained the upper bound of the sum of normalized Fourier coefficients(?)In 2001,Ivic[18]considered the sum of normalized Fourier coefficients over squares,(?)where A are proper positive constants.Subsequently,Fomenko[9]improved Ivic's result(?)For any holomorphic cusp form of even integral k satisfying k?1/x3(logx)22/3,Sankaranarayanan showed that(?)Later for any ?>0,Lu[28]proved(?)In addition,Rankin[32]and Selberg[34]proved(?)In 2009,Lao and Sankaranarayanan[23]obtained the estimate(?)where j=2,3,4.Let Sr denote the set of normalized primitive Maass cusp forms of eigenvalue?=1/4+r2.Then the Fourier expansion of f? Sr is(?)where ?f(n)? R are eigenvalues of Tn and the K-Bessel function Ks(y)is defined aS(?)for y>0 and s? C.Up to now,the best estimate proved by Kim and Sarnak[22]iS|?f(n)|?n7/64d(n).In 1989,Hafner and Ivic[11]proved(?)In 2011,Lau and Lii[25]studied the distribution of ?f(n)over squares,(?)For 1 ?j?4,Jiang and Lii[16]proved that(?)where (?).The asymptotic formula of the sum of ?f2(nj)over arithmetic progressions has been firstly proved by Andrianov and Fomenko[2]for holomorphic cusp form.Later Akbarov[1]improved the error term further.Then,for f? Hk or f?Sr,Jiang and Lu[15]considered the sum of ?f2j(n)over arithmetic progressions for j= 2,3,4.In this paper,for j=2,3,4,as x??,we will discuss the sum of ?f2(nj)over arithmetic progressions,namely,(?)where g ? Z with 0 ?l<q and(q,l)=1.The main results are as follows.Theorem 1 Let f ?q is a prime and(q,l)=1.For any ?>0,if q?x0j,j=2,3,4,then we have(?)where Rj(x,q)=(?) Cj are constants,Fj(s,x0)=(?)(?)Theorem 2 Let f? Sr,q is a prime and(q,l)=1.For any ?>0,j=2,3,4,if q?x,then we have(?)where Rj(x,q)=(?)Dj are constants,Fj(s,x0)(?)...