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

Abstract

Using a different approach, we derive integral representations for the Riemann zeta function and its generalizations (the Hurwitz zeta, $\zeta(-k,b)$, the polylogarithm, $\mathrm{Li}_{-k}(e^m)$, and the Lerch transcendent, $\Phi(e^m,-k,b)$), that coincide with their Abel-Plana expressions. A slight variation of the approach leads to different formulae. We also present the relations between each of these functions and their partial sums. It allows one to figure, for example, the Taylor series expansion of $H_{-k}(n)$ about $n=0$ (when $k$ is a positive integer, we obtain a finite Taylor series, which is nothing but the Faulhaber formula). The method used requires evaluating the limit of $\Phi\left(e^{2\pi i\,x},-2k+1,n+1\right)+\pi i\,x\,\Phi\left(e^{2\pi i\,x},-2k,n+1\right)/k$ when $x$ goes to $0$, which in itself already makes for an interesting problem.