Many odd zeta values are irrational

Stéphane Fischler,Johannes Sprang,Wadim Zudilin

doi:10.1112/s0010437x1900722x

Abstract

Building upon ideas of the second and third authors, we prove that at least$2^{(1-\unicode[STIX]{x1D700})(\log s)/(\text{log}\log s)}$values of the Riemann zeta function at odd integers between 3 and$s$are irrational, where$\unicode[STIX]{x1D700}$is any positive real number and$s$is large enough in terms of$\unicode[STIX]{x1D700}$. This lower bound is asymptotically larger than any power of$\log s$; it improves on the bound$(1-\unicode[STIX]{x1D700})(\log s)/(1+\log 2)$that follows from the Ball–Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.

Highlights

When s 2 is an even integer, the value ζ(s) of the Riemann zeta function is a non-zero rational multiple of πs and, a transcendental number. No such relation is expected to hold for ζ(s) when s 3 is odd; a folklore conjecture states that the numbers π, ζ(3), ζ(5), ζ(7), . . . are algebraically independent over the rationals
The crucial point of this construction is that each ζ(i) appears in this Q-linear combination with a coefficient that depends on d in a very simple way
We assume that 1 j D − 1 and we prove that for any k and any the internal sum over i in Equation (2.4) is an integer

Summary

Introduction

When s 2 is an even integer, the value ζ(s) of the Riemann zeta function is a non-zero rational multiple of πs and, a transcendental number. T=k which is a Q-linear combination of 1 and odd zeta values starting from ζ(5), Rivoal has proved [Riv02] that among the numbers ζ(5), ζ(7), . The crucial point of this construction is that each ζ(i) appears in this Q-linear combination with a coefficient that depends on d in a very simple way This makes it possible to eliminate from the entire collection of these linear combinations as many odd zeta values as the number of divisors of D. Taking D equal to a power of 2 and s sufficiently large with respect to D, the second author proves that at least log D/log 2 numbers are irrational among ζ(3), ζ(5), .

Construction of linear forms

Asymptotic estimates of the linear forms

Elimination of odd zeta values