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

TL;DR

This paper introduces a novel integral representation approach for the Riemann zeta function and related functions, providing insights into their properties and series expansions, applicable across the entire complex plane.

Contribution

It presents new integral formulas for the zeta and related functions, aligning with Abel-Plana expressions, and explores their relations and series expansions, including finite Taylor series for specific cases.

Findings

Derived integral representations valid in all complex numbers

Established relations between functions and their partial sums

Connected series expansions to Faulhaber's formula

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.