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=λt(n)f,where the Hecke operators is(?)Let Hk*be the set of Hecke primitive eigencuspforms of weight k for Γ=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,λf(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|λf(n)|≤d(n),where d(n)is the divisor function.The first important upper estimate about the Hecke eigenvalue λf(n)was ob-tained by E.Hecke[9]in 1927,and showed that(?)In 1990,Rankin[25]proved that.(?) where 0<δ<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 λf2(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∑λf4(n2)=xP2(logx)+O(x79/81+ε),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 λf(n)denote the Hecke eigenvalue.Then for any ε>0,we have(?),n≤x where P5(x)is a polynomial in x of degree 5.Theorem 2.Let f∈Hk*,and λsymjf(n)denote the coefficients of the j-th symmetric power L-functions.Then for any ε>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 λf2(n5),and obtain the following result.Theorem 3.Let f∈Hk*,and λf(n)denote the Hecke eigenvalue.Then we have∑λf2(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 λf(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 φ(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 λf(n)is the Hecke eigenvalue,and obtain the ij≤x following theorem.Theorem 4.Let k G N,q=2.4,6,8,then(ⅰ)for any Riemann integrable function f:[0,1]→R,the following equality holds lim(?)(ⅱ)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,(ⅲ)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.