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

TL;DR

This paper introduces a quick, distributional approach using Cesàro limits to derive the classical values of the Riemann zeta function at negative integers, offering a concise alternative to traditional methods.

Contribution

It presents a novel derivation of zeta at negative integers using Cesàro limits of distributions, simplifying the classical proofs.

Findings

Derived (-n)=-rac{B_{n+1}}{n+1} for positive integers n

Discussed the derivative '( ext{alpha}) of the zeta function

Computed '(0) value

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 Ces\`aro 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 \[ \zeta(-n)=-\frac{B_{n+1}}{n+1}, \] for $n\in\mathbb{N}^+.$ We also give a short discussion on $\zeta^{\prime }(\alpha)$ and compute the value of $\zeta^{\prime}(0)$.