## Statistics for Riemann zeta zeros

• The following table was produced by Agatha L. Lee as part of an undergraduate project using the data linked here.

The table below contains summary statistics about the Hardy $Z$-function (the rotated zeta function) based on data obtained using the amortized algorithm. The 4th column displays the minimum (resp. maximum) normalized spacing of consecutive zeros of the $Z$-function for each dataset of 10 million zeros at heights $T =$ 1e12,...,1e28. The zero ordinate where the max (resp. min) normalized spacing occurs is expressed in the form $T+t$, where $T$ is given in the 2nd column and $t$ is given in the 3rd column. The 5th column displays the maximum of the $Z$-function between the consecutive zero ordinates where the max (resp. min) normalized spacing occurs. The derivatives of the $Z$-function at the consecutive zeros where the max (resp. min) normalized spacing occurs are shown in the 6th and 7th columns.
Summary statistics for the $Z$-function at various heights $T$
Row # $T$ $t$ Normalized.Spacing Max.between.consecutive.zeros Derivative.Zeros.left Derivative.Zeros.right
1 1e12 1690521.47416274 0.00460625 0.00007948 -0.28329158 0.28339492
2 1e12 206880.85325403 3.77365656 165.64638502 43.94693787 -24.91982357
3 1e13 253301.81124073 0.00552191 0.00029029 -0.93922876 0.94132176
4 1e13 1302815.08231513 3.73937893 251.30113621 -101.65507669 45.27135084
5 1e14 389993.89907488 0.00407607 0.00002887 -0.13699606 0.13712265
6 1e14 2026158.53608729 3.96710353 223.91026313 44.34761533 -76.01636763
7 1e15 677902.54443196 0.00348682 0.00001511 0.09025437 -0.09016508
8 1e15 742072.67642370 3.61295524 106.65130623 97.20378372 -25.93228269
9 1e16 381734.48714284 0.00360112 0.00009780 0.60661266 -0.60377489
10 1e16 1267101.54021076 3.73479966 117.22057915 -24.96801799 68.24640454
11 1e17 419484.86315535 0.00241857 0.00000519 0.05095856 -0.05096247
12 1e17 296191.94631050 3.68651602 305.36683221 -157.74261074 242.41183067
13 1e18 941064.90769908 0.00569272 0.00005120 -0.22619900 0.22739615
14 1e18 1246399.50185418 3.72077811 74.91980616 -22.48846801 23.35120343
15 1e19 460628.14464702 0.00540438 0.00001095 0.05400934 -0.05414715
16 1e19 578940.59483533 3.83964540 53.28273741 -22.25117391 16.08472455
17 1e20 1383900.88093245 0.00462024 0.00003397 0.20755877 -0.20632409
18 1e20 368718.16305013 3.84760292 483.50645996 -400.52110929 205.90560850
19 1e21 959614.40713735 0.00398210 0.00005492 0.40809162 -0.40867713
20 1e21 1104477.79613337 3.80635523 151.26620010 -56.01444379 114.99716592
21 1e22 680742.43467852 0.00218283 0.00000709 0.10090500 -0.10111871
22 1e22 48475.43844621 3.83366769 376.08174575 -312.16775699 243.03472023
23 1e23 56641.20788677 0.00401394 0.00001603 0.12983660 -0.13012480
24 1e23 523760.82226342 3.67153966 581.55602221 -566.42704945 635.74403848
25 1e24 792252.53051333 0.00621630 0.00003088 -0.16903572 0.16883176
26 1e24 710245.73832348 3.57551078 401.35098140 98.85842501 -821.65451449
27 1e25 870877.33724656 0.00234681 0.00000979 -0.14786651 0.14815679
28 1e25 209560.46958613 4.09737216 341.03621261 -177.74551811 130.58175584
29 1e26 1050812.35659205 0.00384856 0.00002620 0.25184501 -0.25108456
30 1e26 676401.29576434 3.68488266 58.43612078 -54.00554017 22.96289208
31 1e27 779334.21581987 0.00252168 0.00000333 -0.05069822 0.05076840
32 1e27 486245.57155325 3.68679489 105.24525813 66.38086481 -51.24806696
33 1e28 289065.69360204 0.00255474 0.00000247 0.03847333 -0.03858979
34 1e28 680627.75794282 3.62400200 699.30021929 -1398.39381919 495.10516602