Computations of the Riemann zeta function

These pages sorted by the size of $t$

These pages sorted by the size of $Z(t)$

These pages sorted by the size of $S(t)$

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.

The images are all links that will take you to a zoomable version of the plot.

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).

See also:

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$\zeta(1/2 + it)$ around $t = 891210424622870710406880313 \approx 8.91210424623 \times 10^{ 26 }$

Largest value of $Z(t)$ in this graph:6661.200376

Value of $t$ for which the maximum occurs:891210424622870710406880333.87015234

Value of $\zeta(1/2 + it)$:$6248.227203 - 2308.949388i$

Maximum of $S(t)$ in this range:2.862778276

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 297212370308751345986703821912 \approx 2.97212370309 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:3086.331841

Value of $t$ for which the maximum occurs:297212370308751345986703821932.36599609

Value of $\zeta(1/2 + it)$:$3011.859561 + 673.903716i$

Maximum of $S(t)$ in this range:-2.861553626

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 70391066310491324308791969554433 \approx 7.03910663105 \times 10^{ 31 }$

Largest value of $Z(t)$ in this graph:-14055.89286

Value of $t$ for which the maximum occurs:70391066310491324308791969554453.24909766

Value of $\zeta(1/2 + it)$:$12667.14879 - 6091.918058i$

Maximum of $S(t)$ in this range:-2.859244495

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 89869857757291582322528149832 \approx 8.98698577573 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:7295.672412

Value of $t$ for which the maximum occurs:89869857757291582322528149852.33799609

Value of $\zeta(1/2 + it)$:$7117.657993 - 1601.805431i$

Maximum of $S(t)$ in this range:-2.855984676

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 11700045041631728214881198897 \approx 1.17000450416 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:2837.836829

Value of $t$ for which the maximum occurs:11700045041631728214881198917.27299609

Value of $\zeta(1/2 + it)$:$-31.84193729 - 2837.658182i$

Maximum of $S(t)$ in this range:2.85520269

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 549927995422277597978521852516 \approx 5.49927995422 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:2952.392224

Value of $t$ for which the maximum occurs:549927995422277597978521852536.17830078

Value of $\zeta(1/2 + it)$:$2248.472863 + 1913.371273i$

Maximum of $S(t)$ in this range:2.852039156

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 6263430629274347712240701152 \approx 6.26343062927 \times 10^{ 27 }$

Largest value of $Z(t)$ in this graph:3840.750052

Value of $t$ for which the maximum occurs:6263430629274347712240701172.49010547

Value of $\zeta(1/2 + it)$:$1332.151225 - 3602.323427i$

Maximum of $S(t)$ in this range:-2.851351612

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 222795271975141839186726115317 \approx 2.22795271975 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:3958.409438

Value of $t$ for which the maximum occurs:222795271975141839186726115337.27532422

Value of $\zeta(1/2 + it)$:$2023.064201 - 3402.383946i$

Maximum of $S(t)$ in this range:2.84961886

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 840254377093000239632447060594 \approx 8.40254377093 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:9474.568549

Value of $t$ for which the maximum occurs:840254377093000239632447060614.47414453

Value of $\zeta(1/2 + it)$:$7616.255739 + 5635.609789i$

Maximum of $S(t)$ in this range:2.847742768

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 303710769550561210185584673592 \approx 3.03710769551 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:5730.804231

Value of $t$ for which the maximum occurs:303710769550561210185584673612.84299609

Value of $\zeta(1/2 + it)$:$4925.149041 - 2930.021169i$

Maximum of $S(t)$ in this range:2.845531359

zeta function picture

zeta function picture zeta function picture