Proportions of the non-trivial zeros of the Riemann zeta function

Abstract

Let $C_0=\frac32-\frac1{\sqrt2}\cot\big(\frac1{\sqrt2}\big) =0.67250\ldots$ and $C_1=\frac{C_0+1}{2}= 0.83625\ldots$. Recently, it is obatained by Alpöge and Furman that more than 67.25\% of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and more than 83.62\% are distinct. Later, Lamzouri gave a different and more direct proof. In this article, by refining the method of Lamzouri, we slightly improve the bounds $C_0$ and $C_1$ in these two results to $C_0+δ_0$ and $C_1+\frac{δ_0}2$ with $δ_0=6.66624\ldots\times10^{-8}$, respectively.