Largest known zero of the Riemann zeta function

Question

Numerical calculations on the zeroes of the Riemann zeta function have reached a very high degree of refinement and sophistication and I think that the first $10^{20}$ (with positive imaginary part) or more have been calculated, all simple and all located on the critical line $\Re s=1/2$.

Question 1: what is the order of magnitude of the largest known zero?

Question 2: assuming RH and writing the $k$-th zero as $z_k=\frac12+i t_k$ with a non-decreasing sequence $t_k$ of positive numbers, is there an asymptotic formula for the size of $t_k$ in terms of $k$?

Question 3: if the Riemann hypothesis is wrong, up to a double logarithmic error, say if in fact, no better approximation than $$ \pi(x)=\text{Li}(x)+O\bigl(x^{1/2}(\ln x)( \ln \ln x)\bigr), $$ holds true, how far do the numerical computations should go to detect that "doubly logarithmic" quantity of zeroes off the critical line? If it is up to $10^{100}$, there is a chance that in the next 30 years a computer can detect a zero off line. If it is of the order of $10^{10000}$, RH won't be disproved by a computer before the sun becomes a red giant.

Answer 1

Regarding Question 1, for their paper The zeta function on the critical line: numerical evidence for moments and random matrix theory models, Hiary and Odlyzko computed 5 billion zeros near the $10^{23}$rd zero. The last had imaginary part approximately $$ 1.30664344087959822199974045053551×10^{22} $$

See Table 2 of http://www.dtc.umn.edu/~odlyzko/doc/zeta.moments.pdf

This seems to be the current record.

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Update: In “Alan Turing and the Riemann Zeta Function” by Hejhal and Odlyzko, in the book Alan Turing - His Work and Impact, Elsevier 2013, they write “It is now known that the RH is true for … some hundreds of zeros near zero number $10^{32}$” (This is $t$ near $9.04808\cdot 10^{30}$.)

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Update March 2023

In New Computations of the Riemann Zeta Function on the Critical Line, Bober and Hiary set a new record, computing zero number $n=10^{36} + 42420637374017961984$ with $\gamma_n\approx 8.10292\cdot 10^{34}$.

Answer 2

For (2), according to OEIS A013629 Floor of imaginary parts of nontrivial zeros of Riemann zeta function $$ t_n \sim \frac{2\pi n}{\log{n}}$$

For (1) probably searching for "siegel z" riemann computation large will give some results.

Answer 3

The largest zeta zero built into Mathematica 8 and apparently also in Wolfram Alpha as of 13.7.2021 is:

ZetaZero[10^7]
N[%]
0.5 + 4.99238*10^6 I

If one is happy with one significant decimal digit then this root function:

(*Mathematica start*)
Clear[f, s, n];
nn = 15;(*10^15 zeta zero*)f[x_] := Zeta[x];
(*The Franca-LeClair approximation of the zeta zeros:*)
n = 6;(*increase "n" for better precision.*)
(*The precision of N[11/8] needs to be increased too, accordingly.*)
s = 1/2 + 
   I*Table[2*Pi*Exp[1]*Exp[ProductLog[(10^n - N[11/8])/Exp[1]]], {n, 
      nn, nn}];
(*Root function for almost any function:*)
Monitor[
 z = Table[
   s[[j]] + 
    1/(1 - Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/
          f[k/n + s[[j]] - 1/n], {k, 1, n}]/
        Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/f[k/n + s[[j]]], {k,
           1, n}]), {j, 1, 1}], j]
Zeta[z]
(*end*)

can compute double that in order of magnitude ($10^{15}$), before the built in Riemann zeta function stops evaluating.

ZetaZero[10^15]
is approximately:
{0.580063 + 2.08514*10^14 I}

The formula is based on a conjectured formula where $s$ is the value from the Franca LeClair approximation for the $j$-th Riemann zeta zero made exact by the limit:

$$s+\lim_{n \rightarrow \infty}\left(\left(1-\frac{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s-\frac{1}{n}\right)}}{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s\right)}}\right)^{-1}\right)$$

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Mathematica code for the $10^7$ zeta zero: https://pastebin.com/tsJeGNs7