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arXiv · 2609.22316

At least one of the three numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ is irrational

Shekhar Suman
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Abstract

In this article, we prove the irrationality of at least one of the three numbers $ζ(5), ζ(7)$ and $ζ(9)$ is irrational. We find linear forms in $1,ζ(5),ζ(7)$ and $ζ(9)$ and prove their arithmetic by Kranttenthaler and Rivoal. We also refine these estimates by using integral quantity $Φ$ involving specific product of prime powers. We use the results of Kranttenthaler, Rivoal and Zudilin to prove that at least one of the three numbers $ζ(5), ζ(7)$ and $ζ(9)$ is irrational.

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Shekhar Suman. 2026-09-15. At least one of the three numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ is irrational. https://arxiv.org/abs/2609.22316

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