巢湖学院学报2025，Vol.27Issue(3)：61-65,5.DOI:10.12152/j.issn.1672-2868.2025.03.007
ζ(2n+1)的积分表示及广义伯努利数
The Integral Representation of ζ(2n+1)and Generalized Bernoulli Numbers
摘要
Abstract
The research objective of this article is to generalize Euler's result,which represents the values of the zeta function at positive even points in closed form using Bernoulli numbers,to the positive odd points of the zeta function.The article establishes an integral relationship between the values of the zeta function at positive odd points and generalized Bernoulli numbers.Moreover,the integral representation of rational linear combinations of ζ(2n+1)/π2n,n≥1 is obtained by using hyperbolic functions.The conclusion of the article extends the connection between the values of the zeta function at positive integer points and(generalized)Bernoulli numbers.At the same time,it establishes a method to represent the values of the zeta function at positive integer points in the form of in-tegrals.关键词
黎曼泽塔函数/积分表示/广义伯努利数/双曲函数Key words
Riemann zeta function/integral representation/generalized Bernoulli numbers/hyperbolic functions分类
数理科学引用本文复制引用
吴亚运..ζ(2n+1)的积分表示及广义伯努利数[J].巢湖学院学报,2025,27(3):61-65,5.基金项目
合肥师范学院高层次人才项目(项目编号:2022rcjj24) （项目编号:2022rcjj24）
