The Asymptotic Distributions Of Hecke Eigenvalues

In recent years,the distribution of Hecke eigenvalue has been a hot issue in the study of number theory.By the study of its nature,it is helpful to solve a series of problems in analytic number t.heory,and further to study other related fields in number theory.In this paper,we study some asymptotic properties of Hecke eigenvalues.Let k be a positive even integer,Sk(r)be the set of cusp forms on SL2(Z),f?Sk(r)be the eigenfunction of Hecke operators,Tnf=?t(n)f,where the Hecke operators is(?)Let Hk*be the set of Hecke primitive eigencuspforms of weight k for ?=SL2(Z).The Fourier expansion of f E Hk*at the cusp oo is (?),and satisfies the multiplicative property(?)where m,n?1 are any integers.Therefore,?f(n)is not only the normalized fourier coefficients at the cusp oo but also the normalized eigenvalue of the Hecke operate Tn,abbreviated Hecke eigenvalues in the text.The Hecke L-function attached to f?Hk*is defined by (?),Re(s)>1.In 1974,Deligne[6]proved the Ramanujan-Petersson conjecture|?f(n)|?d(n),where d(n)is the divisor function.The first important upper estimate about the Hecke eigenvalue ?f(n)was ob-tained by E.Hecke[9]in 1927,and showed that(?)In 1990,Rankin[25]proved that.(?) where 0<?<0.06.Finally,in 2009,Wu[31]proved the bast result to date is(?)In 1940,Rankin[26]and Selberg[28]invented the powerful Rankin-Selberg method to study the average behavior of ?f2(n),and showed that(?)In 2012.Lao[17]investigated the fourth moment of Hecke eigenvalue of f E Hk*over square numbers,and showed that??f4(n2)=xP2(logx)+O(x79/81+?),n?x where P2(x)is a polynomial in x of degree 2.Since Xsymjf(n)is closely related to Asymj(n),it is easy to get(?),n?x where P2(x)is a polynomial in x of degree 2.In 2015,Dieulefait[5]deduced that sym5f is automorphic on SL2(Z).In Chapter 2,according to the properties of symmetric power L-functions and their Rankin-Selberg L-functions,we study the average estimation of Hecke eigenvalue on sequences {n2},and obtain the following results.Theorem 1.Let f ? Hk*,and ?f(n)denote the Hecke eigenvalue.Then for any ?>0,we have(?),n?x where P5(x)is a polynomial in x of degree 5.Theorem 2.Let f?Hk*,and ?symjf(n)denote the coefficients of the j-th symmetric power L-functions.Then for any ?>0,we have(?),n?x where P5(x)is a polynomial in x of degree 5.In 2014,Lao[18]made some research on the average estimation of Hecke eigen-value in sparse integer sequences,and showed the following asymptotic formula(?),n?x where j=2,3,4.Therefore,in Chapter 2,according to Dieulefait[5],and using Landau lemma(Lemma 2.7),we study the mean value of ?f2(n5),and obtain the following result.Theorem 3.Let f?Hk*,and ?f(n)denote the Hecke eigenvalue.Then we have??f2(n5)=c1x+O(x35/37),n?x where c1 is a suitable constant.Recently,Lii[15]-[16]further studied the higher moments of the Hecke eigen-value ?f(n),and proved that(?)where Pi(x)is a polynomial of degree i in x.In 2015,Banescu and Popa[4]studied some asymptotics of (?),here ?(n)is the Euler function and d(n)is the divisor ij?x ij?*function.Inspired by these properties,in Chapter 3.we also study some asymptotics for double sums (?)which ?f(n)is the Hecke eigenvalue,and obtain the ij?x following theorem.Theorem 4.Let k G N,q=2.4,6,8,then(?)for any Riemann integrable function f:[0,1]?R,the following equality holds lim(?)(?)if w:[0,1]?R is Riemann integrable,v1:[0,1]?·[0,1],…,vk:[0,1]?[0,1]are all continuous functions,with v1(1)…vk(1)?0,the following equality holds lim (?)dt,(?)if f0:[0,1]?R is Riemann integrable,v1:[0,1]?R,…,vk:[0,1]?R are all continuous functions,with v1(1)…uv(1)?0,the following equality holds lim (?)…vk(?)dt,where Pl(x)is a polynomial of degree l in x,and if q=2,then l=1;if g=4,then l=3;if q=6,then l=9;if q=8,then 1=27.