Special Values for the Riemann Zeta Function

作     者：John H. Heinbockel John H. Heinbockel

作者机构：Old Dominion University Norfolk Virginia USA 

出 版 物：《Journal of Applied Mathematics and Physics》 (应用数学与应用物理（英文）)

年 卷 期：2021年第9卷第5期

页      面：1108-1120页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Riemann Zeta Function Zeta (2n) Zeta (2n + 1) Apéry’s Constant Catalan Constant 

摘      要：The purpose for this research was to investigate the Riemann zeta function at odd integer values, because there was no simple representation for these results. The research resulted in the closed form expression for representing the zeta function at the odd integer values 2n+1 for n a positive integer. The above representation shows the zeta function at odd positive integers can be represented in terms of the Euler numbers E2n and the polygamma functions ψ(2n)(3/4). This is a new result for this study area. For completeness, this paper presents a review of selected properties of the Riemann zeta function together with how these properties are derived. This paper will summarize how to evaluate zeta (n) for all integers n different from 1. Also as a result of this research, one can obtain a closed form expression for the Dirichlet beta series evaluated at positive even integers. The results presented enable one to construct closed form expressions for the Dirichlet eta, lambda and beta series evaluated at odd and even integers. Closed form expressions for Apéry’s constant zeta (3) and Catalan’s constant beta (2) are also presented.