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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Page 2


$\zeta(1/2 + it)$ around $t = 9178358656494989336431259004785 \approx 9.17835865649 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:89.10334759

Value of $t$ for which the maximum occurs:9178358656494989336431259004813.13432422

Value of $\zeta(1/2 + it)$:$88.02846135 - 13.79842543i$

Maximum of $S(t)$ in this range:-1.72348387

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 6632378187823588974002457910686 \approx 6.63237818782 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:12021.09404

Value of $t$ for which the maximum occurs:6632378187823588974002457910706.59637891

Value of $\zeta(1/2 + it)$:$11940.69306 - 1388.002644i$

Maximum of $S(t)$ in this range:2.812395579

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 6083028695276545807063248346835 \approx 6.08302869528 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:9739.335524

Value of $t$ for which the maximum occurs:6083028695276545807063248346855.90626953

Value of $\zeta(1/2 + it)$:$9646.371908 - 1342.447564i$

Maximum of $S(t)$ in this range:-2.992678907

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 5866392475614912729251488020197 \approx 5.86639247561 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:-4423.29945

Value of $t$ for which the maximum occurs:5866392475614912729251488020217.85599609

Value of $\zeta(1/2 + it)$:$1510.595869 - 4157.364327i$

Maximum of $S(t)$ in this range:-2.713054828

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 4051424673556972085158532092501 \approx 4.05142467356 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:1808.875889

Value of $t$ for which the maximum occurs:4051424673556972085158532092521.90899609

Value of $\zeta(1/2 + it)$:$990.9912509 - 1513.264128i$

Maximum of $S(t)$ in this range:-2.816553916

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 3962231703483297661331732963213 \approx 3.96223170348 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:-7484.299572

Value of $t$ for which the maximum occurs:3962231703483297661331732963233.07137109

Value of $\zeta(1/2 + it)$:$6553.017273 - 3615.619546i$

Maximum of $S(t)$ in this range:-2.73733147

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 3805475614378624044873696329590 \approx 3.80547561438 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:4548.845784

Value of $t$ for which the maximum occurs:3805475614378624044873696329610.07149609

Value of $\zeta(1/2 + it)$:$3262.806677 - 3169.556838i$

Maximum of $S(t)$ in this range:2.814192511

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 3746873807855083274300801372785 \approx 3.74687380786 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:-3600.809016

Value of $t$ for which the maximum occurs:3746873807855083274300801372805.47503516

Value of $\zeta(1/2 + it)$:$2037.143416 + 2969.15346i$

Maximum of $S(t)$ in this range:-2.63778651

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 3260615873269180792527357738396 \approx 3.26061587327 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:-274.1037064

Value of $t$ for which the maximum occurs:3260615873269180792527357738416.28599609

Value of $\zeta(1/2 + it)$:$273.3937962 - 19.71481739i$

Maximum of $S(t)$ in this range:-1.622825415

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 2409972808744819410276834556260 \approx 2.40997280874 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:-9268.208332

Value of $t$ for which the maximum occurs:2409972808744819410276834556280.21999609

Value of $\zeta(1/2 + it)$:$8950.064468 - 2407.494905i$

Maximum of $S(t)$ in this range:-2.587162053

zeta function picture

zeta function picture zeta function picture