Discrete mean square of the coefficients of symmetric square L-functions on certain sequence of positive numbers

被引：0

作者：

Anubhav Sharma

Ayyadurai Sankaranarayanan

机构：

[1] University of Hyderabad Central University,School of Mathematics and Statistics

来源：

Research in Number Theory | 2022年 / 8卷

关键词：

Cauchy–Schwarz inequality; Symmetric square ; -function; Holomorphic cusp forms; Principal Dirichlet character; 11M; 11M06;

D O I：

暂无

中图分类号：

学科分类号：

摘要：

In this paper, we will be concerned with the average behavior of the nth\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\mathrm{th}$$\end{document} normalized Fourier coefficients of symmetric square L-function (i.e., L(s,sym2f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(s,sym^{2}f)$$\end{document}) over certain sequence of positive integers. Precisely, we prove an asymptotic formula for ∑a2+b2+c2+d2≤x(a,b,c,d)∈Z4λsym2f2(a2+b2+c2+d2),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathop {\sum }\limits _{\begin{array}{c} a^{2}+b^{2}+c^{2}+d^{2}\le {x} \\ (a,b,c,d)\in {\mathbb {Z}}^{4} \end{array}}\uplambda ^{2}_{sym^{2}f}(a^{2}+b^{2}+c^{2}+d^{2}), \end{aligned}$$\end{document}where x≥x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge {x_{0}}$$\end{document} (sufficiently large), and L(s,sym2f):=∑n=1∞λsym2f(n)ns.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} L(s,sym^{2}f):= \mathop {\sum }\limits _{n=1}^{\infty }\dfrac{\uplambda _{sym^{2}f}(n)}{n^{s}}. \end{aligned}$$\end{document}

引用

共 20 条

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[12] Sankaranarayanan A(1989)Zeros of principal l-functions and random matrix theory Compos. Math. 70 245-130
[13] Rudnick Z(1975)Zeros of quadratic zeta-functions on the critical line Proc. Lond. Math. Soc. 3 79-3876
[14] Sarnak P(2013)On certain l-functions Arch. Math. 100 123-undefined
[15] Sankaranarayanan A(2013)Third symmetric power l-functions for gl(2) J. Number Theory 133 3862-undefined
[16] Shahidi F(undefined)On the holomorphy of certain Dirichlet series undefined undefined undefined-undefined
[17] Shahidi F(undefined)Estimates for the Fourier coefficients of symmetric square l-functions undefined undefined undefined-undefined
[18] Shimura G(undefined)Average behavior of Fourier coefficients of cusp forms over sum of two squares undefined undefined undefined-undefined
[19] Tang H(undefined)undefined undefined undefined undefined-undefined
[20] Zhai S(undefined)undefined undefined undefined undefined-undefined

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