On the first simultaneous sign change for Fourier coefficients associated with Hecke–Maass forms

被引：0
作者：
Guodong Hua
机构：
[1] Shandong University,School of Mathematics
来源：
The Ramanujan Journal | 2022年 / 59卷
关键词：
Hecke–Maass forms; Fourier coefficients; Sign changes; Primary 11F41; Secondary 11F30;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
Let f and g be two distinct Hecke–Maass cusp forms of weight zero for SL(2,Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL(2,\mathbb {Z})$$\end{document} with Laplacian eigenvalues 14+u2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{4}+u^{2}$$\end{document} and 14+v2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{4}+v^{2}$$\end{document}, respectively. Let λf(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{f}(n)$$\end{document} and λg(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{g}(n)$$\end{document} be the real normalized Fourier coefficients satisfies λf(1)=λg(1)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{f}(1)=\lambda _{g}(1)=1$$\end{document}. Then in this paper we give a quantitative result concerning the sign changes of {λf(n)λg(n)}n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\lambda _{f}(n)\lambda _{g}(n)\}_{n\in \mathbb {N}}$$\end{document} in a short interval. We refine the results given by Kumari and Sengupta (Ramanujan J 55:205–218, 2021) concerning the first sign change of the sequence {λf(n)λg(n)}n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\lambda _{f}(n)\lambda _{g}(n)\}_{n\in \mathbb {N}}$$\end{document}.
引用
收藏
页码：559 / 570
页数：11
相关论文
共 49 条
[11]  
Kohnen W(2009)On the Euler products and the classification of automorphic forms II Proc. Am. Math. Soc. 137 3563-3567
[12]  
Sengupta J(2018)Cuspidality of symmetric power with applications Int. J. Number Theory 14 2291-2301
[13]  
Jacquet H(2021)On the signs of Fourier coefficients of cusp forms Ramanujan J. 55 205-218
[14]  
Shalika JA(2021)Signs of Fourier coefficients of two cusp forms of different weights Sci. China Math. 64 1-14
[15]  
Jacquet H(2010)Simultaneous non-vanishing and sign changes of Fourier coefficients of modular forms Number theory. Ser. Number Theory Appl. 6 184-216
[16]  
Shalika JA(2019)The first simultaneous sign change for Fourier coefficients of Hecke-Maass forms Acta Math. Hungar. 159 287-298
[17]  
Kim H(2012)On the first negative Hecke eigenvalue of an automorphic representation of Proc. Camb. Philos. Soc. 152 207-222
[18]  
Shahidi F(2017)A quick introduction to Maass forms Proc. Am. Math. Soc. 145 563-575
[19]  
Knopp M(2010)Some problems involving Hecke eigenvalues Publ. Math. Inst. Hautes Études Sci 111 171-271
[20]  
Kohnen W(1982)On signs of Fourier coefficients of cusp forms Ann. Math. Pura Appl. 130 287-306
12345