A note on the number of irrational odd zeta values

It is proved that, for all odd integer $s⩾s_0(\epsilon )$, there are at least $(c_0-\epsilon )\frac{s^{1\mathrm{/}2}}{(\log s{)}^{1\mathrm{/}2}}$ many irrational numbers among the following odd zeta values: $\zeta (3),\zeta (5),\zeta (7),\cdots ,\zeta (s)$. The constant $c_0 = 1.192507\ldots$ 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.