华东师范大学学报（自然科学版） >

2020 , Vol. 2020 >Issue 6: 1 - 15

DOI: https://doi.org/10.3969/j.issn.1000-5641.201911023

数学

乘法群$ {\mathbb{Z}}^{\times}(m)$上的三角函数的傅里叶变换

  • 沈力健
展开
  • 佛罗里达大学 数学系, 盖恩斯维尔 FL 32611-8105, 美国
沈力健, 男, 教授, 研究方向为函数论. E-mail: shen@ufl.edu

收稿日期: 2019-05-21

  网络出版日期: 2020-12-01

收起

The Fourier transform of trigonometric functions on the multiplicative group ${\mathbb Z}^{\times}(m)$

  • Lichien SHEN
Expand
  • Department of Mathematics, University of Florida, Gainesville FL 32611-8105, USA

Received date: 2019-05-21

  Online published: 2020-12-01

Fold

摘要

根据乘法群上的傅里叶变换理论框架, 研究了一类三角和, 并揭示了这类三角和与许多数论量 (例如高斯和、 虚二次域类数和伯努利数) 之间的有趣联系.

关键词： 特征; 傅里叶变换; 高斯和; 克罗内克符号; theta函数

本文引用格式

沈力健 . 乘法群$ {\mathbb{Z}}^{\times}(m)$上的三角函数的傅里叶变换[J]. 华东师范大学学报（自然科学版）, 2020 , 2020(6) : 1 -15 . DOI: 10.3969/j.issn.1000-5641.201911023

Abstract

Based on the Fourier transform on the multiplicative group $ {\mathbb Z}^{\times}(m)$, we study a class of trigonometric sums and reveal interesting connections between these sums and number theoretic quantities, such as Gauss sums, the class number of imaginary quadratic fields, and the Bernoulli number.

Key words： character; Fourier transform; Gauss sum; Kronecker symbol; theta function

参考文献

[1] BERNDT B C, ZHANG L C. A new class of theta-function identities originating in Ramanujan’s notebook [J]. J Number Theory, 1994, 48: 224-242
[2] LIU Z G. Some Eisenstein series identities related to modular equation of seventh order [J]. Pacific J Math, 2003, 209: 103-130
[3] BOREVICH Z I, SHAFAREVICH I R. Number Theory [M]. New York: Academic Press, 1966.
[4] ERDELYI A. Higher Transcendental Functions [M]. New York: McGraw-Hill, 1953.
[5] SHEN L C. On the products of three theta functions [J]. Ramanujan J, 1999(3): 343-357
Options
文章导航

/

〈 〉