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Small improvements on the Ball-Rivoal theorem and its $p$-adic variant

Li Lai

TL;DR

This work sharpens the Ball–Rivoal theme on the linear independence of $1$ and odd zeta values by tightly integrating Zudilin’s $oldsymbol{igPhi}_n$ denominator method. It yields explicit, non-vanishing linear forms in $1$ and odd zeta values, establishing a near-universal lower bound on the dimension of their $Q$-span for large even $s$, and provides a parallel $p$-adic result for $zeta_p(s)$ with an analogous constant. The approach remains explicit and programming-free for the main bounds, while computational parameter tuning delivers a sharper constant of approximately $1.119/(1+ ext{log }2)$ in a broader setting. Complementing this, the paper offers detailed $p$-adic preliminaries via Volkenborn integrals and overconvergent power series, together with a carefully constructed p-adic analogue of Nesterenko’s criterion and asymptotics. The results enhance the understanding of the arithmetic nature of zeta values, deliver explicit non-vanishing linear forms, and extend the Ball–Rivoal framework to the $p$-adic realm with explicit, quantitative bounds.

Abstract

We prove that the dimension of the $\mathbb{Q}$-linear span of $1,ζ(3),ζ(5),\ldots,ζ(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s$. This slightly refines a well-known result of Rivoal (2000) or Ball-Rivoal (2001). Quite unexpectedly, the proof only involves inserting the arithmetic observation of Zudilin (2001) into the original proof of Ball-Rivoal. Although this result is covered by a recent development of Fischler (2021+), our proof has the advantages of being simple and providing explicit non-vanishing small linear forms in $1$ and odd zeta values. Moreover, we establish the $p$-adic variant: for any prime number $p$, the dimension of the $\mathbb{Q}$-linear span of $1,ζ_p(3),ζ_p(5),\ldots,ζ_p(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s$. This is new, it slightly refines a result of Sprang (2020).

Small improvements on the Ball-Rivoal theorem and its $p$-adic variant

TL;DR

This work sharpens the Ball–Rivoal theme on the linear independence of and odd zeta values by tightly integrating Zudilin’s denominator method. It yields explicit, non-vanishing linear forms in and odd zeta values, establishing a near-universal lower bound on the dimension of their -span for large even , and provides a parallel -adic result for with an analogous constant. The approach remains explicit and programming-free for the main bounds, while computational parameter tuning delivers a sharper constant of approximately in a broader setting. Complementing this, the paper offers detailed -adic preliminaries via Volkenborn integrals and overconvergent power series, together with a carefully constructed p-adic analogue of Nesterenko’s criterion and asymptotics. The results enhance the understanding of the arithmetic nature of zeta values, deliver explicit non-vanishing linear forms, and extend the Ball–Rivoal framework to the -adic realm with explicit, quantitative bounds.

Abstract

We prove that the dimension of the -linear span of is at least for any sufficiently large even integer . This slightly refines a well-known result of Rivoal (2000) or Ball-Rivoal (2001). Quite unexpectedly, the proof only involves inserting the arithmetic observation of Zudilin (2001) into the original proof of Ball-Rivoal. Although this result is covered by a recent development of Fischler (2021+), our proof has the advantages of being simple and providing explicit non-vanishing small linear forms in and odd zeta values. Moreover, we establish the -adic variant: for any prime number , the dimension of the -linear span of is at least for any sufficiently large even integer . This is new, it slightly refines a result of Sprang (2020).
Paper Structure (27 sections, 32 theorems, 288 equations, 1 figure, 2 tables)

This paper contains 27 sections, 32 theorems, 288 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Notations
  3. An outline of the proof of Theorem \ref{['mainthm']}
  4. Similarities between the $p$-adic case and the classical case
  5. Structure of the paper
  6. Linear independence criterion
  7. Setting of the proof for the classical case
  8. Parameters
  9. Rational functions
  10. Linear forms
  11. Arithmetic properties for the classical case
  12. Asymptotic estimates for the classical case
  13. Proof of Theorem \ref{['mainthm']}
  14. Preliminaries for $p$-adic zeta values
  15. Overconvergent power series on $\mathbb{Z}_p$
  16. ...and 12 more sections

Key Result

Theorem 1

We have as the odd integer $s \to +\infty$.

Figures (1)

  • Figure 1: the values of $\phi(x,y)$

Theorems & Definitions (65)

  • Theorem : Rivoal Riv2000, 2000; Ball-Rivoal BR2001, 2001
  • Theorem : Sprang Spr20, 2020
  • Theorem 1.1
  • Theorem 1.2
  • Claim 1.3
  • Claim 1.4
  • Theorem : FischlerFis2021+, 2021+
  • Theorem 2.1: Nesterenko's linear independence criterion Nes1985, 1985
  • Theorem 2.2: Fischler-Zudilin FZ2010, 2010
  • Theorem 2.3: Nesterenko's $p$-adic linear independence criterion Nes2012, 2012
  • ...and 55 more