PreprintCC BY 4.0arXiv
At least one of the three numbers ζ(5), ζ(7), ζ(9) is irrational
Abstract
In this article, we prove the irrationality of at least one of the three numbers $ζ(5), ζ(7)$ and ζ(9) is irrational. We find linear forms in $1,ζ(5),ζ(7)$ and ζ(9) and prove their arithmetic by Kranttenthaler and Rivoal. We also refine these estimates by using integral quantity Φ involving specific product of prime powers. We use the results of Kranttenthaler, Rivoal and Zudilin to prove that at least one of the three numbers $ζ(5), ζ(7)$ and ζ(9) is irrational.
Related
- Motzkin Numbers Count 2-Stack-Sortable Permutations Ending in Their Least Entry
- Totally Geodesic Submanifolds of Teichmüller Space in Complex Dimension at Least Three
- A density deficit for sums of three cube-full numbers
- On the lower- and upper bounds of three-colored anti-Ramsey numbers
- Harmonic Vector Fields and Betti Numbers in Bounded Three-Dimensional Electromagnetic Domains
BibTeX
@misc{suman2026least,
title = {{At least one of the three numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ is irrational}},
author = {Shekhar Suman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22316},
eprint = {2609.22316},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22316},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - At least one of the three numbers ζ(5), ζ(7), ζ(9) is irrational
AU - Shekhar Suman
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.22316
UR - https://arxiv.org/abs/2609.22316
AB - In this article, we prove the irrationality of at least one of the three numbers $ζ(5), ζ(7)$ and ζ(9) is irrational. We find linear forms in $1,ζ(5),ζ(7)$ and ζ(9) and prove their arithmetic by Kranttenthaler and Rivoal. We also refine these estimates by using integral quantity Φ involving specific product of prime powers. We use the results of Kranttenthaler, Rivoal and Zudilin to prove that at least one of the three numbers $ζ(5), ζ(7)$ and ζ(9) is irrational.
ER -