November 9, 2025Open Access

Perspectives on the arithmetic nature of the ratios ζ (2n + 1) /π^2n+1 and β (2n) /π^2n

Key Points

Irrationality of ζ(2n + 1)/π^{2n+1} and β(2n)/π^{2n} is suggested by the investigation.
Key focus includes values of the Riemann zeta function at odd integers and their relationships.
The analysis unites distinct analytic frameworks and presents conjectures that enhance understanding of these mathematical functions.
Insights gained may lead to significant implications for number theory and mathematical irrationality.

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 ζ (2n + 1) /π^2n+1 and β (2n) /π^2n

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Cite This Study

Luc Ramsès Talla Waffo (Wed,) studied this question.

synapsesocial.com/papers/690fdcdaf60c54d04ea37fdc https://doi.org/https://doi.org/10.48550/arxiv.2511.02843