对称幂平方L-函数Fourier系数的高次均值估计

Mean Value Estimates for Higher Moments of Fourier Coefficients of Symmetric Square L-functions

设f(z)为全模群SL(2,Z)上权为偶数k的Hecke特征形式,L(s,sym^(2)f)为其对应的对称幂平方L-函数,λsym^(2)f(n)为对称幂平方L-函数L(s,sym^(2)f)的Fourier展开式中第n个正规化Fourier系数.本文借助Newton和Thorne近期关于函数Symjf的重要工作以及自守L-函数的亚凸界和均值估计结果,研究了λsym^(2)f(n)在四个整数平方和序列上的三次均值估计和四次均值估计,得到了均值估计的渐近公式,改进了之前的结果.

Let f(z)be a Hecke eigenform of even weight k for the full modular group SL(2,Z),L(s,sym^(2)f)is the symmetric square L-function associated of f.Denote byλsym^(2)f(n)the nth normalized Fourier coefficient of the Fourier expansion of symmetric square L-function L(s,sym^(2)f).In this paper,we establish asymptotic formulas for the third moment ofλsym^(2)f(n)and the fourth moment ofλsym^(2)f(n)over a sequence of sum of squares of four integers.We improve previous results and our improvement benefit from the recent work of Newton and Thorne in function Sym j f and subconvex bounds and mean value estimates of the automorphic L-function.