PreprintCC0arXiv
A linear independence criterion for p-adic Hurwitz zeta values at odd integers
Abstract
In this article, a new criterion is established to obtain the linear independence of some values of the p-adic Hurwitz zeta function at odd integers s≥ 3. We generalize a theorem of F.~Beukers, who gave, among others, an alternative proof---using Padé approximation theory---of a result originally due to F.~Calegari on the irrationality of the p-adic Riemann zeta value ζp(3) for p = 2, 3. To prove our main theorem, we work in a general setting, where we construct and study new Padé approximants of Laurent series satisfying the functional equation f(z) = f(1−z). This complements our previous work on the linear independence of p-adic polygamma values.
Related
- Linear independence of values of Dirichlet L functions
- The threshold for linear independence of multiple zeta values in positive characteristic
- Linear independence of values of polylogarithms with periodic coefficients, outside the disk of convergence
- Large values of the Hurwitz zeta function with rational parameter
- Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function
BibTeX
@misc{kawashima2026linear,
title = {{A linear independence criterion for $p$-adic Hurwitz zeta values at odd integers}},
author = {Makoto Kawashima and Anthony Poëls},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15525},
eprint = {2610.15525},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15525},
note = {CC0},
}
RIS
TY - UNPB
TI - A linear independence criterion for p-adic Hurwitz zeta values at odd integers
AU - Makoto Kawashima
AU - Anthony Poëls
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15525
UR - https://arxiv.org/abs/2610.15525
AB - In this article, a new criterion is established to obtain the linear independence of some values of the p-adic Hurwitz zeta function at odd integers s≥ 3. We generalize a theorem of F.~Beukers, who gave, among others, an alternative proof---using Padé approximation theory---of a result originally due to F.~Calegari on the irrationality of the p-adic Riemann zeta value ζ_p(3) for p = 2, 3. To prove our main theorem, we work in a general setting, where we construct and study new Padé approximants of Laurent series satisfying the functional equation f(z) = f(1−z). This complements our previous work on the linear independence of p-adic polygamma values.
ER -