A note on the number of irrational odd zeta values, II

TL;DR

The paper proves that for sufficiently large even $s$, at least $1.284579 \cdot \sqrt{\frac{s}{\log s}}$ of the odd zeta values $\zeta(i)$ with $3 \le i \le s-1$ are irrational. It achieves this by combining the Fischler-Sprang-Zudilin elimination technique with Zudilin's $\Phi_n$-factor method, building on and surpassing prior irrationality-density results. A parameterized construction of rational functions $R_n(t)$ yields linear forms in Hurwitz zeta values, whose arithmetic and asymptotic properties are controlled via $\rho_{n,i}$, $\rho_{n,0,\theta}$, and $S_{n,\theta}$. The elimination step reduces the problem to a computable optimization: with carefully chosen $(M,J,\delta_j)$ and a table of $\delta_j$, the bound is improved to the explicit constant $C_0=1.284579\ldots$, achieved for $M=563$, $J=76$ and the given $\delta_j$.

Abstract

We prove that there are at least $1.284 \cdot \sqrt{s/\log s}$ irrational numbers among $ζ(3)$, $ζ(5)$, $ζ(7)$, $\ldots$, $ζ(s-1)$ for any sufficiently large even integer $s$. This result improves upon the previous finding by a constant factor. The proof combines the elimination technique of Fischler-Sprang-Zudilin (2019) with the $Φ_n$ factor method of Zudilin (2001).

Theorems & Definitions (26)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof