Riemann Zeta函数的级数表示和一些有趣的恒等式
A Series Representation for Riemann's Zeta Function and some Interesting Identities that Follow
摘要：
本文以柯西积分定理为基础，用复宗量的指数积分函数$E_{s}(iκ)$，得到了Dirichlet函数$η(s)$的一种新的级数表示，从而得到了Riemann函数$ζ(s)$的一种新的级数表示.从这个基础上，无限的总和进行评估，不寻常的积分减少到已知的功能和有趣的身份出土。定义了不完全函数$ζ^{\pm}(s)$和$η^{\pm}(s)$，并证明了它们与某些有趣的积分密切相关.发展了一个关于欧拉数、伯努利数和调和数的恒等式。它表明，一个已知的简单的积分与复杂的端点可以用来评估大量的不同的积分，通过选择不同的端点之间的路径。
英文摘要：
Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $η(s)$, and hence Riemann's function $ζ(s)$, is obtained in terms of the Exponential Integral function $E_{s}(iκ)$ of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions $ζ^{\pm}(s)$ and $η^{\pm}(s)$ are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
