Hecke特征值相邻极大值的均值估计

Mean Value Estimate of Pairwise Maxima of Hecke Eigenvalues

取f为权为偶数k的全模群Γ=SL(2,Z)的Hecke特征型.定义λ_(sym^(m)f)(n)为与f关联的m阶对称幂次L-函数的Dirichlet展开式的第n个正规化系数.本文中,我们给出了∑_(n≤x) max{|λ_(sym^(m)f)(n)|^(2β),|λ_(sym^(m)f)(n+h)|^(2β)}的上下界,其中h为一个固定的正整数,β> 0为一个正数.

Let f be a Hecke eigenform of even integral weight k for the full modular groupΓ=SL(2,Z).Denote by λ_(sym^(m)f)(n) the-n-th normalized coefficient of the Dirichlet expansion of the m-th symmetric power L-function associated to f.In this paper,we establish the bounds for ∑_(n≤x) max{|λ_(sym^(m)f)(n)|^(2β),|λ_(sym^(m)f)(n+h))|^(2β)},where h is a fixed positive integer,and β> 0 is a positive number.