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).

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$\zeta(1/2 + it)$ around $t = 1500000000000000000000000003000 \approx 1.5 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:58.92508758

Value of $t$ for which the maximum occurs:1500000000000000000000000003013.39599609

Value of $\zeta(1/2 + it)$ : $28.72706915 + 51.44824044i$

Maximum of $S(t)$ in this range:1.920177045

$\zeta(1/2 + it)$ around $t = 1500000000000000000000000002000 \approx 1.5 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:249.404057

Value of $t$ for which the maximum occurs:1500000000000000000000000002007.088996094

Value of $\zeta(1/2 + it)$ : $147.878016 + 200.8344493i$

Maximum of $S(t)$ in this range:1.858733602

$\zeta(1/2 + it)$ around $t = 1500000000000000000000000001000 \approx 1.5 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:-82.82322625

Value of $t$ for which the maximum occurs:1500000000000000000000000001029.30549609

Value of $\zeta(1/2 + it)$ : $70.97317311 + 42.69069576i$

Maximum of $S(t)$ in this range:1.922620206

$\zeta(1/2 + it)$ around $t = 1500000000000000000000000000000 \approx 1.5 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:97.26948693

Value of $t$ for which the maximum occurs:1500000000000000000000000000016.25019922

Value of $\zeta(1/2 + it)$ : $75.91258094 - 60.81638875i$

Maximum of $S(t)$ in this range:-1.843742121

$\zeta(1/2 + it)$ around $t = 1420608056968699501169509003459 \approx 1.42060805697 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:5051.118925

Value of $t$ for which the maximum occurs:1420608056968699501169509003479.90644922

Value of $\zeta(1/2 + it)$ : $4523.466648 - 2247.676996i$

Maximum of $S(t)$ in this range:-2.908613306

$\zeta(1/2 + it)$ around $t = 1141639706784284971550986463600 \approx 1.14163970678 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:-5711.799989

Value of $t$ for which the maximum occurs:1141639706784284971550986463620.52244141

Value of $\zeta(1/2 + it)$ : $1083.032682 + 5608.181463i$

Maximum of $S(t)$ in this range:2.938800053

$\zeta(1/2 + it)$ around $t = 1096785418585585487051643762992 \approx 1.09678541859 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:4578.232528

Value of $t$ for which the maximum occurs:1096785418585585487051643763012.23999609

Value of $\zeta(1/2 + it)$ : $-253.9673593 - 4571.18296i$

Maximum of $S(t)$ in this range:-3.039497868

$\zeta(1/2 + it)$ around $t = 949298829754554964058786559858 \approx 9.49298829755 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:-91.24195793

Value of $t$ for which the maximum occurs:949298829754554964058786559875.37699609

Value of $\zeta(1/2 + it)$ : $72.08571989 + 55.93517566i$

Maximum of $S(t)$ in this range:1.696628187

$\zeta(1/2 + it)$ around $t = 949298829754554930008909071187 \approx 9.49298829755 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:-108.5077419

Value of $t$ for which the maximum occurs:949298829754554930008909071201.46619141

Value of $\zeta(1/2 + it)$ : $85.23001621 - 67.15485383i$

Maximum of $S(t)$ in this range:1.586003426

$\zeta(1/2 + it)$ around $t = 945714596951858489397711072913 \approx 9.45714596952 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:-7789.094657

Value of $t$ for which the maximum occurs:945714596951858489397711072933.93114453

Value of $\zeta(1/2 + it)$ : $7486.01452 + 2151.646386i$

Maximum of $S(t)$ in this range:2.944654374