Instant Evaluation and Demystification of zeta(n),L(n,chi) that Euler,Ramanujan Missed III

TL;DR

This paper provides a new integral-based method to evaluate Hurwitz and Riemann zeta functions at specific points, explaining their polynomial nature and explicit values at even integers, contrasting previous power series approaches.

Contribution

It introduces an integration approach to analyze zeta functions, clarifying their polynomial form at non-positive arguments and explicit evaluations at positive even integers.

Findings

Hurwitz zeta function becomes a polynomial at non-positive values

Riemann zeta function can be explicitly evaluated at positive even integers

Integration approach offers new insights over power series methods

Abstract

We show that for a non-positive value of the first variable,Hurwitz zeta function becomes a polynomial in the second variable. We show this, using 'integration approach', instead of 'power series approach', which we had resorted to, in our earlier paper with the same title. This, in particular, explains why Riemann zeta function at positive even integer arguments, can be evaluated and why it cannot be evaluated explicitly at positive odd integer arguments.