Density and distributions of those numerically or analytically KNOWN solutions of Riemann $\zeta(1/2 + r i)=0?$

Question

We know the conjecture about the Riemann hypothesis is about the nontrivial zeros are on $$(1/2 + r i)$$ for some $r \in \mathbb{R}$ of Riemann zeta function.

My question is how much is known about the density and the distributions of those numerically or analytically KNOWN solutions of $$\zeta(1/2 + r i)=0?$$

I found a related post but it was about 8 years ago, so maybe we have a better update?

Mean density of the nontrivial zeros of the Riemann zeta function

Answer 1

In my humble opinion, a key paper is the one published in year $2014$ by G.Franca and A.LeClair. In particular, they provide a very good and simple approximation (equation $(229)$ in the linked paper). $$\Im\left(r _{n}\right) \sim \frac{2 \pi \left(n-\frac{11}{8}\right)}{W\left(\frac{n-\frac{11}{8}}{e}\right)}$$ where $W(.)$ is Lambert function;

Repeating some of their calculations for $n=10^k$, we have $$\left( \begin{array}{ccc} k & \text{approximation} & \text{solution} \\ 1 & 50.233653 & 49.773832 \\ 2 & 235.98727 & 236.52423 \\ 3 & 1419.5178 & 1419.4225 \\ 4 & 9877.6296 & 9877.7827 \\ 5 & 74920.891 & 74920.827 \\ 6 & 600269.64 & 600269.68 \end{array} \right)$$

Answer 2

Mathematica 8.0.1 derivation of Eric Weisstein's approximation for Gram points:

(*Start*)
(*Mathematica*)
(*The derivation of the Gram points approximation by Weisstein in \
Mathworld*)
Clear[x, n, a, g, t];
Series[RiemannSiegelTheta[x], {x, Infinity, 12}]
a = Normal[Series[RiemannSiegelTheta[x], {x, Infinity, 0}]]
g = FullSimplify[(x /. Solve[a == (n)*Pi, x])[[1]]]
n = Range[42] - 2;
t = N[g, 20]
Zeta[1/2 + I*t]
(*End*)

9.6769067871658668471,
17.847836512849620314,
23.171660819240722718,
27.671198036307304064,
31.718791394674873194,
35.467863110275089697,...

Modified Mathematica 8.0.1 derivation of Eric Weisstein's approximation giving Franca-LeClair points:

(*Start*)
(*Mathematica*)
(*Analogous to the derivation of the Gram points approximation by \
Weisstein in Mathworld*)
Clear[x, n, a, g, t];
Series[RiemannSiegelTheta[x], {x, Infinity, 12}]
a = Normal[Series[RiemannSiegelTheta[x], {x, Infinity, 0}]]
g = FullSimplify[(x /. Solve[a == (n + 1/2)*Pi, x])[[1]]]
n = Range[42] - 2;
t = N[g, 20]
Zeta[1/2 + I*t]
(*End*)

14.521346953065628168,
20.655740355699557203,
25.492675432264310733,
29.739411632309551244,
33.624531888500487851,
37.257370086972976394,...

The basic difficulty in getting an accurate asymptotic for the Riemann zeta zeros is that the Riemann-Siegel theta function is not invertible. User reuns pointed out to me that the exact asymptotic for the Riemann zeta zeros has been known for about 120 years and the exact asymptotic is the functional inverse of the Riemann-Siegel theta function, according to the French Wikipedia.