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arXiv · 2207.04013

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

Jose Risomar Sousa
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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 $\Phi\left(e^{2\pi i \,x},-2k+1,n+1\right)+\pi i \,x\,\Phi\left(e^{2\pi i \,x},-2k,n+1\right)/k$ when $x$ goes to zero, which is an interesting problem in itself.

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Jose Risomar Sousa. 2022-07-07. Simplest Integrals for the Zeta Function and its Generalizations Valid in All $\mathbb{C}$. https://arxiv.org/abs/2207.04013

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