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

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.


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