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:

Page 0  Page 1  Page 2  Page 3  Page 4  Page 5  Page 6  Page 7  Page 8  Page 9  Page 10  Page 11  Page 12  Page 13  Page 14  Page 15  Page 16  Page 17  Page 18  Page 19  Page 20  Page 21  Page 22

Page 9


$\zeta(1/2 + it)$ around $t = 171207393801570900621968363457 \approx 1.71207393802 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:5228.432689

Value of $t$ for which the maximum occurs:171207393801570900621968363477.05702734

Value of $\zeta(1/2 + it)$:$4681.757712 + 2327.585254i$

Maximum of $S(t)$ in this range:2.636463167

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 167460505621937453717737789992 \approx 1.67460505622 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:-5095.493177

Value of $t$ for which the maximum occurs:167460505621937453717737790012.39124609

Value of $\zeta(1/2 + it)$:$1939.875535 - 4711.78667i$

Maximum of $S(t)$ in this range:2.716642068

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 155022509712772372546593011496 \approx 1.55022509713 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:-833.2591909

Value of $t$ for which the maximum occurs:155022509712772372546593011498.406996094

Value of $\zeta(1/2 + it)$:$72.59070899 - 830.0912409i$

Maximum of $S(t)$ in this range:-2.200180905

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 151430114521478793791897636280 \approx 1.51430114521 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:9171.086741

Value of $t$ for which the maximum occurs:151430114521478793791897636299.94044141

Value of $\zeta(1/2 + it)$:$8746.666937 - 2757.652897i$

Maximum of $S(t)$ in this range:-2.933250462

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 150000000000000000000000000000 \approx 1.5 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:112.2148424

Value of $t$ for which the maximum occurs:150000000000000000000000000001.737996094

Value of $\zeta(1/2 + it)$:$19.54040032 - 110.5004236i$

Maximum of $S(t)$ in this range:-1.855361979

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 146938392544594231809171188638 \approx 1.46938392545 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:8612.136976

Value of $t$ for which the maximum occurs:146938392544594231809171188658.37899609

Value of $\zeta(1/2 + it)$:$8484.468484 - 1477.395627i$

Maximum of $S(t)$ in this range:-3.01622673

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 139909165720312176640219432458 \approx 1.3990916572 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:-7927.955564

Value of $t$ for which the maximum occurs:139909165720312176640219432478.19906641

Value of $\zeta(1/2 + it)$:$7389.347921 - 2872.284236i$

Maximum of $S(t)$ in this range:-2.704498136

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 133648977500290800142661619768 \approx 1.336489775 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:-8895.692825

Value of $t$ for which the maximum occurs:133648977500290800142661619788.58704297

Value of $\zeta(1/2 + it)$:$8605.44346 - 2253.817581i$

Maximum of $S(t)$ in this range:-2.67494708

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 119436902627122039941399691338 \approx 1.19436902627 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:3577.619333

Value of $t$ for which the maximum occurs:119436902627122039941399691358.71844141

Value of $\zeta(1/2 + it)$:$591.7468084 + 3528.341793i$

Maximum of $S(t)$ in this range:2.660070397

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 117469312393254571652414861471 \approx 1.17469312393 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:7825.657273

Value of $t$ for which the maximum occurs:117469312393254571652414861491.12899609

Value of $\zeta(1/2 + it)$:$7812.166121 + 459.3171566i$

Maximum of $S(t)$ in this range:2.739302715

zeta function picture

zeta function picture zeta function picture