Recurrence Relations for Values of the Riemann Zeta Function in Odd Integers

TL;DR

This paper develops recurrence relations involving the values of the Riemann zeta function at odd integers, providing explicit series representations for the coefficients, advancing understanding of these special values.

Contribution

It introduces new recurrence relations for odd zeta values and explicitly computes series representations for their coefficients, extending known relations for even zeta values.

Findings

Derived recurrence relations for zeta(2k+1) values

Explicit series representations for recurrence coefficients

Enhanced understanding of odd zeta function values

Abstract

It is commonly known that $\zeta(2k) = q_{k}\frac{\zeta(2k + 2)}{\pi^2}$ with known rational numbers $q_{k}$. In this work we construct recurrence relations of the form $\sum_{k = 1}^{\infty}r_{k}\frac{\zeta(2k + 1)}{\pi^{2k}} = 0$ and show that series representations for the coefficients $r_{k} \in \mathbb{R}$ can be computed explicitly.