It is proved that, for all odd integer $s \geqslant s_0(\varepsilon)$, there are at least $\big( c_0 - \varepsilon \big) \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 $c_0 = 1.192507\ldots$ can be expressed in closed form. The work is based on the previous work of Fischler, Sprang and Zudilin [FSZ19], improves the lower bound $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ 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.

中文翻译：

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关于无理奇 zeta 值数量的注释

证明，对于所有奇整数$s \geqslant s_0(\varepsilon)$，至少有$\big( c_0 - \varepsilon \big) \frac{s^{1/2}}{(\log s)^{1/2}} $ 下列奇数 zeta 值中的许多无理数：$\zeta(3),\zeta(5),\zeta(7),\cdots,\zeta(s)$。常数 $c_0 = 1.192507\ldots$ 可以用封闭形式表示。该工作基于 Fischler、Sprang 和 Zudilin [FSZ19] 之前的工作，改进了其中的下界 $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$。主要的新成分是辅助有理函数零点的优化设计，它与欧拉托函数的逆相关。