The generalized Bernoulli numbers and its relation with the Riemann zeta function at odd-integer arguments

TL;DR

This paper introduces new integral representations of the Riemann zeta function at odd integers using generalized Bernoulli numbers, providing criteria for the rational span of specific zeta values over pi.

Contribution

It develops explicit formulas connecting generalized Bernoulli numbers with zeta at odd integers, advancing understanding of their rational relations.

Findings

Derived new integral representations for ζ(2n+1)

Established criteria for the dimension of rational span of ζ(2n+1)/π^{2n}

Enhanced understanding of the algebraic relations among zeta values

Abstract

By using the generalized Bernoulli numbers, we deduce new integral representations for the Riemann zeta function at positive odd-integer arguments. The explicit expressions enable us to obtain criteria for the dimension of the vector space spanned over the rational by the $\zeta(2n+1)/\pi^{2n}$, $n\geq1$.