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

Let $C_0=\frac{3}{2}-\frac{1}{\sqrt{2}}\cot \left(\frac{1}{\sqrt{2}}\right)=0.67250\dots$ and $C_1=\frac{C_0+1}{2}=0.83625\dots$. 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 $C_0+{\delta }_0$ and $C_1+\frac{{\delta }_0}{2}$ with ${\delta }_0=6.66624\dots \times {10}^{-8}$, respectively.