Finite-part integral representation of the Riemann zeta function at odd positive integers and consequent representations

TL;DR

This paper introduces a novel finite-part integral representation for the Riemann zeta function at odd positive integers, leading to new integral formulas and relations involving derivatives of zeta values.

Contribution

It presents a new finite-part integral approach to represent z(2n+1) and derives related integral formulas for their derivatives, advancing understanding of these special values.

Findings

Derived finite-part integral representations for z(2n+1)

Established relations between z(2n+1) and z'(2n+1)

Obtained new integral representations for derivatives of zeta at odd integers

Abstract

The values of the Riemann zeta function at odd positive integers, $\zeta(2n+1)$, are shown to admit a representation proportional to the finite-part of the divergent integral $\int_0^{\infty} t^{-2n-1} \operatorname{csch}t\,\mathrm{d}t$. Integral representations for $\zeta(2n+1)$ are then deduced from the finite-part integral representation. Certain relations between $\zeta(2n+1)$ and $\zeta'(2n+1)$ are likewise deduced, from which integral representations for $\zeta'(2n+1)$ are obtained.