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 $t = 803625728592344363123814218758 \approx 8.03625728592 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:10282.64968

Value of $t$ for which the maximum occurs:803625728592344363123814218778.19937891

Value of $\zeta(1/2 + it)$:$10017.94308 + 2318.124414i$

Maximum of $S(t)$ in this range:2.920400992

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 20167521863315855715464294057 \approx 2.01675218633 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:7471.985684

Value of $t$ for which the maximum occurs:20167521863315855715464294077.03499609

Value of $\zeta(1/2 + it)$:$7450.305881 + 568.7814586i$

Maximum of $S(t)$ in this range:-2.919826113

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 1420608056968699501169509003459 \approx 1.42060805697 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:5051.118925

Value of $t$ for which the maximum occurs:1420608056968699501169509003479.90644922

Value of $\zeta(1/2 + it)$:$4523.466648 - 2247.676996i$

Maximum of $S(t)$ in this range:-2.908613306

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 23250719692201127177747536545 \approx 2.32507196922 \times 10^{ 28 }$

Largest value of $Z(t)$ in this graph:7500.573929

Value of $t$ for which the maximum occurs:23250719692201127177747536565.20433984

Value of $\zeta(1/2 + it)$:$7482.749871 + 516.781996i$

Maximum of $S(t)$ in this range:2.895462974

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 879454993987734164753812769184 \approx 8.79454993988 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:5578.638522

Value of $t$ for which the maximum occurs:879454993987734164753812769204.33599609

Value of $\zeta(1/2 + it)$:$5467.905565 + 1105.99118i$

Maximum of $S(t)$ in this range:-2.891402879

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 214034968914410911693730173700 \approx 2.14034968914 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:-8441.704689

Value of $t$ for which the maximum occurs:214034968914410911693730173720.94199609

Value of $\zeta(1/2 + it)$:$7721.448482 - 3411.980567i$

Maximum of $S(t)$ in this range:-2.890878717

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 9832284408046499500622869540111 \approx 9.83228440805 \times 10^{ 30 }$

Largest value of $Z(t)$ in this graph:-10137.11421

Value of $t$ for which the maximum occurs:9832284408046499500622869540131.74599609

Value of $\zeta(1/2 + it)$:$10097.50605 - 895.2407846i$

Maximum of $S(t)$ in this range:-2.881891872

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 78028717949218948398892829 \approx 7.80287179492 \times 10^{ 25 }$

Largest value of $Z(t)$ in this graph:-5381.325141

Value of $t$ for which the maximum occurs:78028717949218948398892849.87417578

Value of $\zeta(1/2 + it)$:$4965.234181 + 2074.875851i$

Maximum of $S(t)$ in this range:-2.873251972

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 13036589919442157190082573984513 \approx 1.30365899194 \times 10^{ 31 }$

Largest value of $Z(t)$ in this graph:6175.066008

Value of $t$ for which the maximum occurs:13036589919442157190082573984533.98030078

Value of $\zeta(1/2 + it)$:$1152.530548 - 6066.556976i$

Maximum of $S(t)$ in this range:2.871716171

zeta function picture

zeta function picture zeta function picture


$\zeta(1/2 + it)$ around $t = 693830873990128577485748288481 \approx 6.9383087399 \times 10^{ 29 }$

Largest value of $Z(t)$ in this graph:8828.179585

Value of $t$ for which the maximum occurs:693830873990128577485748288501.64839453

Value of $\zeta(1/2 + it)$:$8810.469972 + 558.9039748i$

Maximum of $S(t)$ in this range:2.871557265

zeta function picture

zeta function picture zeta function picture