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

TL;DR

This paper develops new power series and rapidly decreasing series expressions for the derivatives of Hurwitz zeta and Riemann zeta functions at specific points, providing insights into classical identities and evaluations at special arguments.

Contribution

It introduces novel series representations for derivatives of zeta functions and unifies proofs of classical identities, advancing analytical techniques in number theory.

Findings

Power series for derivatives of Hurwitz zeta at non-positive integers

Rapidly decreasing series for Riemann zeta at odd integers

Unified proof of classical zeta identities

Abstract

For Hurwitz zeta function, we obtain power series expression in second variable for its higher order derivatives (with respect to first variable) at non-positive integer arguments and consequently obtain rapidly decreasing series expression for Riemann zeta fuction at positive odd integer arguments. Further, we obtain corresponding results for Dirichlet L-series. We also a unified proof of various classical identities involving Riemann zeta values.