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:

- Ghaith Hiary's similar description of these computation.
- Slides (with some typos and bad mistakes) from a talk I gave on this stuff at MSRI in April 2011. Paper coming one day.
- The LMFDB, which should contain this data some day.

$\zeta(1/2 + it)$ around $t = 1436161885496321078553725637 \approx 1.4361618855 \times 10^{ 27 }$

Largest value of $Z(t)$ in this graph:873.6024904

Value of $t$ for which the maximum occurs:1436161885496321078553725657.08512891

Value of $\zeta(1/2 + it)$:$381.5845992 + 785.8590871i$

Maximum of $S(t)$ in this range:-2.01736117

$\zeta(1/2 + it)$ 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

$\zeta(1/2 + it)$ around $t = 134032020307222475497920429 \approx 1.34032020307 \times 10^{ 26 }$

Largest value of $Z(t)$ in this graph:513.8356594

Value of $t$ for which the maximum occurs:134032020307222475497920449.44699609

Value of $\zeta(1/2 + it)$:$287.522093 + 425.8616336i$

Maximum of $S(t)$ in this range:2.003195135

$\zeta(1/2 + it)$ around $t = 134032020307222475497920429 \approx 1.34032020307 \times 10^{ 26 }$

Largest value of $Z(t)$ in this graph:513.8356538

Value of $t$ for which the maximum occurs:134032020307222475497920449.44699609

Value of $\zeta(1/2 + it)$:$287.5220899 + 425.8616289i$

Maximum of $S(t)$ in this range:2.003191169

$\zeta(1/2 + it)$ 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

$\zeta(1/2 + it)$ 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

$\zeta(1/2 + it)$ around $t = 10000000000000000000000000920 \approx 1.0 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:263.1295123

Value of $t$ for which the maximum occurs:10000000000000000000000000928.548996094

Value of $\zeta(1/2 + it)$:$212.8902123 + 154.6444237i$

Maximum of $S(t)$ in this range:1.747703799

$\zeta(1/2 + it)$ 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 $t = 10000000000000000000000000680 \approx 1.0 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:167.7062212

Value of $t$ for which the maximum occurs:10000000000000000000000000710.67737891

Value of $\zeta(1/2 + it)$:$163.3093428 + 38.15016595i$

Maximum of $S(t)$ in this range:1.820897073

$\zeta(1/2 + it)$ around $t = 16000000000000000000000000120 \approx 1.6 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:159.8677628

Value of $t$ for which the maximum occurs:16000000000000000000000000137.47948047

Value of $\zeta(1/2 + it)$:$-75.73409037 - 140.7907992i$

Maximum of $S(t)$ in this range:-2.6040471