Many p-Adic Odd Zeta Values Are Irrational
Abstract
For any prime p and ε>0, we prove that for sufficiently large positive odd integers s, at least (cp−ε) s logs of the p-adic zeta values ζp(3),ζp(5),…,ζp(s) are irrational. The constant cp is positive and does only depend on p. This result establishes a p-adic version of the elimination technique used by Fischler, Sprang, and Zudilin and by Lai and Yu to prove a similar result on classical zeta values. The main difficulty consists in proving the nonvanishing of the resulting linear forms. We overcome this problem by using a new irrationality criterion.
Fields of Study & Subjects
Advanced Mathematical Identities
Analytic Number Theory Research
Algebraic Geometry and Number Theory
Publication details
- Journal / source
- The Michigan Mathematical Journal
- Volume
- -1
- Issue
- -1
- Pages
- 1–36
- Publication date
- 1 January 2026
- Language
- EN