A new upper bound for the irrationality exponent of $ζ(5)$

Abstract

Fauzan's determinantal proof that $ζ(5)$ is irrational also yields $|ζ(5)-a/b|>b^{-260}$ for all large $b$. The exponent is the product $\varrhoλ/\varepsilon$ of three constants attached to the construction, of which $\varrho$ governs the growth of a relative matrix norm. We show that $\varrho$ may be taken to be $9.09$ instead of $6933/500=13.866$, and deduce $|ζ(5)-a/b|>b^{-171}$ for all integers $a$ and all sufficiently large integers $b$, so that the irrationality exponent of $ζ(5)$ is at most $171$. Two closed-form functions replace two worst-case estimates: a node function, which bounds the residue weights individually instead of uniformly, and the external field of the energy estimate, which replaces a factor-by-factor bound on the weight. The result is conditional on Fauzan's preprint. An exact arithmetic verifier is attached.