Simplest Integrals for the Zeta Function and its Generalizations Valid in All $\mathbb{C}$

Abstract

A formula for the Riemann zeta function is obtained as an extension of the Faulhaber formula and, from there, formulae for its generalizations (the Hurwitz zeta, $\zeta (-k,b)$, the polylogarithm, ${\mathrm{Li}}_{-k}(e^z)$, and the Lerch transcendent, $\Phi (e^z,-k,b)$), are derived that coincide with their Abel-Plana expressions. It allows one to find, for example, the Taylor series expansion of $H_{-k}(n)$ about $n=0$ (when $k$ is a positive integer, a finite Taylor series is obtained, which is nothing but the Faulhaber formula). The used method requires evaluating the limit of $\Phi \left(e^{2\pi ix},-2k+1,n+1\right)+\pi ix\Phi \left(e^{2\pi ix},-2k,n+1\right)/k$ when $x$ goes to zero, which is an interesting problem in itself.

arXiv commentInaccurate claims about the domain have been rectified