Does the Euler constant allow to go from the prime realm to the zeta zeros realm?

Question

On average the $n$-th critical zero of the Riemann zeta function has imaginary part around $\frac{2\pi n}{\log n}$, so that the average gap between the $n$-th and $n+1$-th zero is $\frac{2\pi}{\log n}$. That way, the reciprocal thereof is the average prime gap $g_{n}:=p_{n+1}-p_{n}$.

Taking into account the critical zeros of zeta of negative imaginary part, computations on wolfram alpha lead to this:

$$\displaystyle{\dfrac{4\pi\sum_{k=1}^{n}\frac{1}{\Im(\rho_{k+1}-\rho_{k})}}{n}-\gamma\approx\dfrac{p_{n+1}-2}{n}+\gamma}$$, where $\gamma$ is the Euler constant.

So my questions are:

1) does the difference between thd LHS and the RHS tend to $0$ as $n$ tends to $\infty$?

2) if yes to 1), does $x\mapsto x+\gamma$ play an analogous role to $g_n\mapsto\frac{2\pi}{g_n}$ as an additive factor to add to go from the prime gap realm to the zeta zero gap realm? Or more philosophically from the discrete realm of integers to the continuous realm of real numbers?

Answer 1

Here is a Mathematica program to plot the LHS and RHS for the 49 first terms of n:

nn = 50
g1 = ListLinePlot[
   Table[4*Pi*Sum[1/(aa[[k + 1]] - aa[[k]]), {k, 1, n}]/n - 
     EulerGamma, {n, 1, nn - 1}]];
g2 = ListLinePlot[
   Table[(Prime[n + 1] - 2)/n + EulerGamma, {n, 1, nn - 1}]];
Show[g1, g2]

And here is a Mathematica program to plot the LHS and RHS for the 10000-1 first terms of n:

nn = 10000
g1 = ListLinePlot[
   Table[4*Pi*Sum[1/(aa[[k + 1]] - aa[[k]]), {k, 1, n}]/n - 
     EulerGamma, {n, 1, nn - 1}]];
g2 = ListLinePlot[
   Table[(Prime[n + 1] - 2)/n + EulerGamma, {n, 1, nn - 1}]];
Show[g1, g2]