I have noticed an interesting property related to the Gibbs phenomenon for the Fourier transform of the zeta zeros in Riemann's explicit formula, namely that the rate at which $r\rightarrow 2 $ in the interval $[2,3]$
where $r$ is the point at which
$$\operatorname{li}(x)-\sum_{\rho}\operatorname{li}(x^\rho)-\log 2+\int_{x}^{\infty}(dt)/(t(t^2-1)\log t)=1$$
for partial sums of $\sum_{\rho}\operatorname{li}(x^\rho).$
From initial observations, it seems that for each sucessive zero added, $r-2\sim C/\operatorname{li}(n)$ for some $C<1/2.$
Much the same results can be achieved with finding $r$ for the partials sums at the points where
$$n-\sum_{\gamma}^{}\dfrac{2\log n\sin(\gamma \log n )}{\gamma\sqrt{n}} = 5/2$$
where $\gamma=$ imaginary parts of zeta zeros.
Does this suggest that the zeta zeros and the prime powers are in some sort of one to one correspondance?
Resposted to MO here