Many odd zeta values are irrational. (English) Zbl 1430.11097

Fischler, Stéphane; Sprang, Johannes; Zudilin, Wadim

Compos. Math. 155, No. 5, 938-952 (2019).

The main result of this paper is that if \(\varepsilon >0\) and \(s\geq 3\) is an odd number sufficiently large with respect to \(\varepsilon\), then among the numbers \(\zeta(3), \zeta(5), \zeta(7),\dots ,\zeta(s)\) at least \[ 2^{\frac{(1-\varepsilon)(\log s)}{\log \log s}} \] are irrational. The proof is based on the suitable estimations of specific linear forms.

Reviewer: Jaroslav Hančl (Ostrava)

MSC:

11J72	Irrationality; linear independence over a field
11M06	\(\zeta (s)\) and \(L(s, \chi)\)
33C20	Generalized hypergeometric series, \({}_pF_q\)

Keywords:

irrationality; zeta values; hypergeometric series; linear forms

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