A Series Representation for Riemann's Zeta Function and some Interesting Identities that Follow

Abstract

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta (s)$, and hence Riemann's function $\zeta (s)$, is obtained in terms of the Exponential Integral function $E_s(i\kappa )$ 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 ${\zeta }^{\pm }(s)$ and ${\eta }^{\pm }(s)$ 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.

arXiv commentComments added to Eqs. (2.32) and (2.46); added new result (2.49) and new reference

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