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The linear independence of $1$, ζ(3), and ζ(4)
Abstract
In a recent breakthrough, Fauzan proved the irrationality of ζ(5). In this paper, we combine Fauzan's determinant method with classical simultaneous approximations of zeta values studied by Zudilin and Dauguet--Zudilin to prove the linear independence of $1$, ζ(3), and ζ(4) over ℚ. As a corollary, ζ(3)/π4 is irrational.
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BibTeX
@misc{lai2026linear,
title = {{The linear independence of $1$, $ζ(3)$, and $ζ(4)$}},
author = {Li Lai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06838},
eprint = {2610.06838},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06838},
}
RIS
TY - UNPB
TI - The linear independence of $1$, ζ(3), and ζ(4)
AU - Li Lai
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.06838
UR - https://arxiv.org/abs/2610.06838
AB - In a recent breakthrough, Fauzan proved the irrationality of ζ(5). In this paper, we combine Fauzan's determinant method with classical simultaneous approximations of zeta values studied by Zudilin and Dauguet--Zudilin to prove the linear independence of $1$, ζ(3), and ζ(4) over ℚ. As a corollary, ζ(3)/π^4 is irrational.
ER -