PreprintCC BY 4.0arXiv
A new upper bound for the irrationality exponent of zeta(3)
Abstract
We prove that the irrationality exponent of ζ(3) satisfies μ(ζ(3)) < 5.5138800, improving the bound $5.513891$ obtained by Rhin and Viola in 2001, which has been the best known for twenty-five years. The proof is an application of their Theorem 5.1 at a new integral direction. That direction is not reached by enlarging the search range: its eight parameters are 52000 times theirs, shifted by at most 23. The mechanism is the one recently used by Bai for π. In particular the Rhin-Viola point, optimal in every region covered by enumeration, is not optimal.
Related
- A new upper bound for the irrationality exponent of ζ(2)
- A new upper bound for the irrationality exponent of ζ(5)
- An Upper Bound to the Undetected Error Exponent with Erasure Decoding at Positive Rates
- Ideal-Smooth Sets and an Explicit Upper Bound for the Real Sum-Product Exponent
- A degree-linear upper bound for the Pólya exponent of forms with fixed arity
BibTeX
@misc{giraudin2026upper,
title = {{A new upper bound for the irrationality exponent of zeta(3)}},
author = {David Niedbala Giraudin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26980},
eprint = {2609.26980},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26980},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A new upper bound for the irrationality exponent of zeta(3)
AU - David Niedbala Giraudin
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26980
UR - https://arxiv.org/abs/2609.26980
AB - We prove that the irrationality exponent of ζ(3) satisfies μ(ζ(3)) < 5.5138800, improving the bound $5.513891$ obtained by Rhin and Viola in 2001, which has been the best known for twenty-five years. The proof is an application of their Theorem 5.1 at a new integral direction. That direction is not reached by enlarging the search range: its eight parameters are 52000 times theirs, shifted by at most 23. The mechanism is the one recently used by Bai for π. In particular the Rhin-Viola point, optimal in every region covered by enumeration, is not optimal.
ER -