Perspectives on the arithmetic nature of the ratios $\zeta(2n + 1)/\pi^{2n+1}$ and $\beta(2n)/\pi^{2n}$

TL;DR

This paper explores the properties of special mathematical constants derived from the Riemann zeta and Dirichlet beta functions, offering new analytic insights and conjectures about their irrationality.

Contribution

It unifies various analytic approaches to these constants and proposes conjectures on their linear independence and irrationality.

Findings

Collected multiple analytic frameworks for these constants

Proposed conjectures on linear independence and irrationality

Provided a unifying perspective on their properties

Abstract

We investigate the values of the Riemann zeta function at odd integers and the Dirichlet beta function at even integers, by collecting several distinct analytic frameworks converging to these values, thus providing a unifying perspective. Beyond analytic interest, these formulas motivate linear independence conjectures which, if established, would imply the irrationality of the quantities $\zeta(2n + 1)/\pi^{2n+1}$ and $\beta(2n)/\pi^{2n}$