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 ϱλ/ε of three constants attached to the construction, of which ϱ governs the growth of a relative matrix norm. We show that ϱ 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.
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BibTeX
@misc{giraudin2026upper,
title = {{A new upper bound for the irrationality exponent of $ζ(5)$}},
author = {David Niedbala Giraudin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01708},
eprint = {2611.01708},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01708},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A new upper bound for the irrationality exponent of ζ(5)
AU - David Niedbala Giraudin
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01708
UR - https://arxiv.org/abs/2611.01708
AB - 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 ϱλ/ε of three constants attached to the construction, of which ϱ governs the growth of a relative matrix norm. We show that ϱ 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.
ER -