L重卷积L函数相关的傅里叶系数和的性质

Properties of Sums of Fourier Coefficients Related to L-Functions of L-Fold Convolutions

摘要: 设$f$ 是全模群$\Gamma =SL\left(2,\mathbb{Z}\right)$ 上的全纯尖点形式，这类形式是所有Hecke算子$T_n$ 的公共本征函数。设$H_k^{\text{*}}$ 表示$\Gamma =SL\left(2,\mathbb{Z}\right)$ 上具有偶整数权$k$ 的标准化本原全纯Hecke尖点形式构成的集合。本文主要关注与一般乘积L-函数相关的系数${\displaystyle \sum _{n\le X}{\lambda }_{f\otimes f\otimes \cdots {\otimes }_{l_1}f}^{t_1}\left(n\right){\lambda }_{g\otimes g\otimes \cdots {\otimes }_{l_2}g}^{t_2}\left(n\right)}$ 的均值，这里${\lambda }_{f\otimes f\otimes \cdots {\otimes }_{l_1}f}^{t_1}\left(n\right)$ 代表$l_1$ 次卷积的$t_1$ 次幂。

Abstract: Let $f$ be a holomorphic cusp form for the full modular group $\Gamma =SL\left(2,\mathbb{Z}\right)$ , which is a simultaneous eigenfunction of all Hecke operators $T_n$ . Let $H_k^{\text{*}}$ denote the set of normalized primitive holomorphic Hecke cusp forms of even integral weight $k$ for $\Gamma =SL\left(2,\mathbb{Z}\right)$ . We focus on the average behavior of coefficients associated with general product L-functions ${\displaystyle \sum _{n\le X}{\lambda }_{f\otimes f\otimes \cdots {\otimes }_{l_1}f}^{t_1}\left(n\right){\lambda }_{g\otimes g\otimes \cdots {\otimes }_{l_2}g}^{t_2}\left(n\right)}$ , the notation ${\lambda }_{f\otimes f\otimes \cdots {\otimes }_{l_1}f}^{t_1}\left(n\right)$ here represents the $t_1$ -th power of the coefficients of the $l_1$ -fold convolution of $f$ .