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 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 ± (s) and η ± (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.