Computations of the Riemann zeta function

Here are some pictures of and information about

$Z(t)$ and

$S(t)$ for some large values of

$t$ . The

$Z$ function is the zeta function on the critical line, rotated so that it is real, so

$Z(t) = e^{i Arg(\zeta(1/2 + it)} \zeta(1/2 + it)$

$S(t)$ is the argument of

$\zeta(1/2 + it)$ , properly interpreted. In some way, it measures irregularity in the distribution of the zeros of the zeta function.

These are from computations run by Ghaith Hiary and myself, based on the algorithm described in Ghaith's paper (also available at the arXiv). These computations have been run on a variety of machines. Initially, we used machines on the Sage cluster at the University of Washington (thanks to William Stein and the NSF), then later the riemann cluster at University of Waterloo (thanks to Mike Rubinstein). Currently, computations are being run at the University of Bristol on the LMFDB machines (funded by EPSRC) and on BlueCrystal.

If your web browser window is big enough, in the top right of each section below you will see a plot of Z(t), in the bottom left you will see S(t), and in the bottom right you will see a zoomed in plot of Z(t). Things are sized roughly so that this looks good on my 1080p monitor.

You can click on any image for a bigger version. Also, you can look at a list of all of the images: Z(t) or S(t).

$\zeta(1/2 + it)$ around $t = 1436161885496321078553725637 \approx 1.4361618855 \times 10^{ 27 }$

Largest value of $Z(t)$ in this graph:873.6024904

Value of $t$ for which the maximum occurs:1436161885496321078553725657.08512891

Value of $\zeta(1/2 + it)$ : $381.5845992 + 785.8590871i$

Maximum of $S(t)$ in this range:-2.01736117

$\zeta(1/2 + it)$ around $t = 134032020307222475497920429 \approx 1.34032020307 \times 10^{ 26 }$

Largest value of $Z(t)$ in this graph:513.8356594

Value of $t$ for which the maximum occurs:134032020307222475497920449.44699609

Value of $\zeta(1/2 + it)$ : $287.522093 + 425.8616336i$

Maximum of $S(t)$ in this range:2.003195135

$\zeta(1/2 + it)$ around $t = 134032020307222475497920429 \approx 1.34032020307 \times 10^{ 26 }$

Largest value of $Z(t)$ in this graph:513.8356538

Value of $t$ for which the maximum occurs:134032020307222475497920449.44699609

Value of $\zeta(1/2 + it)$ : $287.5220899 + 425.8616289i$

Maximum of $S(t)$ in this range:2.003191169

$\zeta(1/2 + it)$ around $t = 10000000000000000000000000640 \approx 1.0 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:-54.15462849

Value of $t$ for which the maximum occurs:10000000000000000000000000656.01299609

Value of $\zeta(1/2 + it)$ : $54.15462729 - 0.01142650706i$

Maximum of $S(t)$ in this range:-1.995594207

$\zeta(1/2 + it)$ around $t = 16000000000000000000000000200 \approx 1.6 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:76.51552714

Value of $t$ for which the maximum occurs:16000000000000000000000000217.67502734

Value of $\zeta(1/2 + it)$ : $73.29971836 - 21.94942328i$

Maximum of $S(t)$ in this range:1.985930957

$\zeta(1/2 + it)$ around $t = 10000000000000000000000000200 \approx 1.0 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:-79.43794806

Value of $t$ for which the maximum occurs:10000000000000000000000000230.06099609

Value of $\zeta(1/2 + it)$ : $71.74138878 + 34.11100597i$

Maximum of $S(t)$ in this range:-1.960355901

$\zeta(1/2 + it)$ around $t = 109990955615748542241920601 \approx 1.09990955616 \times 10^{ 26 }$

Largest value of $Z(t)$ in this graph:-57.22646044

Value of $t$ for which the maximum occurs:109990955615748542241920629.19799609

Value of $\zeta(1/2 + it)$ : $52.04489144 + 23.79489546i$

Maximum of $S(t)$ in this range:1.943736068

$\zeta(1/2 + it)$ around $t = 10000000000000000000000000560 \approx 1.0 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:-81.2537058

Value of $t$ for which the maximum occurs:10000000000000000000000000585.41337109

Value of $\zeta(1/2 + it)$ : $62.40004784 - 52.04227834i$

Maximum of $S(t)$ in this range:1.940107741

$\zeta(1/2 + it)$ around $t = 10000000000000000000000000280 \approx 1.0 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:63.3587181

Value of $t$ for which the maximum occurs:10000000000000000000000000294.37643359

Value of $\zeta(1/2 + it)$ : $63.34350966 + 1.388143794i$

Maximum of $S(t)$ in this range:1.938954282

$\zeta(1/2 + it)$ around $t = 98297762869274426683196572871 \approx 9.82977628693 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:-115.3268952

Value of $t$ for which the maximum occurs:98297762869274426683196572908.90499609

Value of $\zeta(1/2 + it)$ : $114.6584263 - 12.39911389i$

Maximum of $S(t)$ in this range:-1.929701413