For all sufficiently large even s, at least 1.284579 sqrt(s/log s) of the numbers ζ(3), ζ(5), ..., ζ(s-1) are irrational, improving the previous record constant of 1.192507.
Infinitely many odd zeta values are irrational. By elementary means
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In this small note, we provide an elementary proof of the fact that infinitely many odd zeta values are irrational. For the first time, this celebrated theorem been proven by Rivoal and Ball--Rivoal. The original proof uses highly non-elementary methods like the saddle-point method and Nesterenko's linear independence criterion. Recently, Zudilin has re-proven a slightly weaker form of his important result that at least one of the odd zeta values $\zeta(5),\zeta(7),\zeta(9)$ and $\zeta(11)$ is irrational, by elementary means. His new main ingredient are certain 'twists by half' of hypergeometric series. Generalizing this to 'higher twists' allows us to give a purely elementary proof of the result of Rivoal and Ball--Rivoal.
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A note on the number of irrational odd zeta values, II
For all sufficiently large even s, at least 1.284579 sqrt(s/log s) of the numbers ζ(3), ζ(5), ..., ζ(s-1) are irrational, improving the previous record constant of 1.192507.