A note on the number of irrational odd zeta values

Abstract

It is proved that, for all odd integer $s⩾s_0(\epsilon )$, there are at least $(c_0-\epsilon )\frac{s^{1/2}}{(\log s{)}^{1/2}}$ many irrational numbers among the following odd zeta values: $\zeta (3),\zeta (5),\zeta (7),\cdots ,\zeta (s)$. The constant $c_0=1.192507\dots$ can be expressed in closed form. The work is based on the previous work of Fischler, Sprang and Zudilin [FSZ19], improves the lower bound $2^{(1-\epsilon )\frac{\log s}{\log \log s}}$ therein. The main new ingredient is an optimal design for the zeros of the auxiliary rational functions, which relates to the inverse of Euler totient funtion.

15 pages, corrected typos, improved the constant 1/10 to about 1.19