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

$\zeta(1/2 + it)$ 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 function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ 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 function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 1500000000000000000000000005000 \approx 1.5 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:-56.48299812

Value of $t$ for which the maximum occurs:1500000000000000000000000005035.98626172

Value of $\zeta(1/2 + it)$:$56.43803979 + 2.25316255i$

Maximum of $S(t)$ in this range:1.914417045

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 16000000000000000000000000280 \approx 1.6 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:-52.9603908

Value of $t$ for which the maximum occurs:16000000000000000000000000285.464371094

Value of $\zeta(1/2 + it)$:$51.60156147 + 11.91980901i$

Maximum of $S(t)$ in this range:1.911026801

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 10000000000000000000000000080 \approx 1.0 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:64.47512928

Value of $t$ for which the maximum occurs:10000000000000000000000000085.331996094

Value of $\zeta(1/2 + it)$:$60.34910751 - 22.69421773i$

Maximum of $S(t)$ in this range:1.910704816

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 10000000000000000000000000240 \approx 1.0 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:-61.44358057

Value of $t$ for which the maximum occurs:10000000000000000000000000249.555996094

Value of $\zeta(1/2 + it)$:$50.11324921 - 35.55243797i$

Maximum of $S(t)$ in this range:1.903679777

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 10000000000000000000000000480 \approx 1.0 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:-111.9397727

Value of $t$ for which the maximum occurs:10000000000000000000000000510.91416016

Value of $\zeta(1/2 + it)$:$24.10761727 + 109.3130161i$

Maximum of $S(t)$ in this range:-1.899419353

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 10000000000000000000000000360 \approx 1.0 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:83.8344958

Value of $t$ for which the maximum occurs:10000000000000000000000000373.98409766

Value of $\zeta(1/2 + it)$:$18.1033582 + 81.85652758i$

Maximum of $S(t)$ in this range:1.898016433

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 10000000000000000000000000120 \approx 1.0 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:-480.5116167

Value of $t$ for which the maximum occurs:10000000000000000000000000139.82914453

Value of $\zeta(1/2 + it)$:$184.8801747 - 443.5208392i$

Maximum of $S(t)$ in this range:-1.890637995

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 9999999999999999999999999980 \approx 1.0 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:-45.62660934

Value of $t$ for which the maximum occurs:9999999999999999999999999985.290347656

Value of $\zeta(1/2 + it)$:$39.99671851 + 21.95563683i$

Maximum of $S(t)$ in this range:-1.884375111

zeta function picture

zeta function picture zeta function picture