论文标题
Riemann Zeta功能的系列表示以及以下一些有趣的身份
A Series Representation for Riemann's Zeta Function and some Interesting Identities that Follow
论文作者
论文摘要
将Cauchy的积分定理作为基础，可能是Dirichlet函数$η（s）$的新系列表示，因此，根据指数积分函数$ e_ {s}（s}（s}（iκ），复杂参数的riemann函数$ζ（s）$。从此基础上，评估无限总和，将异常的积分简化为已知功能，并发掘了有趣的身份。定义了不完整的函数$ζ^{\ pm}（s）$和$η^{\ pm}（s）$与这些有趣的积分中的一些密切相关。开发了与Euler，Bernouli和谐波数相关的身份。已经证明，可以利用具有复杂端点的已知简单积分来通过选择端点之间的不同路径来评估大量不同的积分。
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.
