A quick distributional way to reproduce some results of the Riemann zeta function

Abstract

The evaluation of the Riemann zeta function at negative integers is a classical result typically obtained through analytic continuation or contour integration. In this paper, we present a novel and concise derivation of these special values by employing the theory of Cesàro limit of distributions, a generalized limit concept developed by Estrada, Kanwal, and Fulling. We use this tool to give a quick proof of the result that \[ ζ(-n)=-\frac{B_{n+1}}{n+1}, \] for $n\in {\mathbb{N}}^{+}.$ We also give a short discussion on ${\zeta }^{\prime }(\alpha )$ and compute the value of ${\zeta }^{\prime }(0)$.