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Computation of zeros of the Zeta function (Gourdon and Demichel, 2004)

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Text183 passages · 0 marked

No examination yetMarked passage
Computation of zeros of the Zeta function0 passages
New Riemann Hypothesis verification record2 passages
1 History of numerical verification of the RH1 passage
Year n Author0 passages
1903 15 J. P. Gram [ 6 ]0 passages
1914 79 R. J. Backlund [ 1 ]0 passages
1925 138 J. I. Hutchinson [ 7 ]0 passages
1935 1,041 E. C. Titchmarsh [ 22 ]0 passages
1953 1,104 A. M. Turing [ 24 ]0 passages
1956 15,000 D. H. Lehmer [ 12 ]0 passages
1956 25,000 D. H. Lehmer [ 11 ]0 passages
1958 35,337 N. A. Meller [ 14 ]0 passages
1966 250,000 R. S. Lehman [ 10 ]1 passage
1977 40,000,000 R. P. Brent [ 2 ]0 passages
1979 81,000,001 R. P. Brent [ 3 ]3 passages
2001 10,000,000,000 J. van de Lune (unpublished)0 passages
2004 900,000,000,000 S. Wedeniwski [ 26 ]7 passages
Close roots2 passages
2 p2 passages
Gram blocks and violations of Rosser rule12 passages
Large values of | Z(t) |1 passage
4 Computations of zeros of the Riemann Hypothesis at very large height2 passages
10 passages
M7 passages
31 passage
451 passage
3157 passages
10 13 33.1 1 2×10 9 167772160 passages
10 14 35.0 3 2×10 9 167772160 passages
10 15 38.3 0 2×10 9 - 1 83886080 passages
10 16 49.5 1 2×10 9 - 1 167772160 passages
10 17 46.9 0 2×10 9 167772160 passages
10 18 81.6 1 2×10 9 - 1 335544320 passages
10 19 65.9 0 2×10 9 +1 335544320 passages
10 20 87.8 4 2×10 9 - 1 335544320 passages
10 21 139.9 0 2×10 9 - 1 335544320 passages
10 22 151.5 2 2×10 9 - 1 1342177280 passages
10 23 219.0 100 2×10 9 - 1 2684354560 passages
10 24 326.6 0 2×10 9 +47 2684354563 passages
M0 passages
{ n : N+1 � n � N+M, d n � [i/100,(i+1)/100]},11 passages
10 13 37.98 68 282468 3.7190 passages
10 14 54.10 86 418346 3.8660 passages
10 15 72.42 109 581126 4.0120 passages
10 16 93.99 140 780549 4.1520 passages
10 17 117.25 171 1004269 4.2830 passages
10 18 142.30 196 1255321 4.4110 passages
10 19 168.55 225 1529685 4.5380 passages
10 20 197.28 270 1837645 4.6570 passages
10 21 225.80 322 2156944 4.7760 passages
10 22 256.53 348 2507825 4.8880 passages
10 23 286.97 480 2868206 4.9970 passages
10 24 319.73 473 3262812 5.1021 passage
As already seen before, it is known unconditionally that1 passage
Under the RH, we have the slightly better bound9 passages
10 13 -2.4979 2.4775 208 237 1.58740 passages
10 14 -2.5657 2.5822 481 411 1.57580 passages
10 15 -2.7610 2.6318 785 760 1.56520 passages
10 16 -2.6565 2.6094 1246 1189 1.55550 passages
10 17 -2.6984 2.6961 1791 1812 1.54650 passages
10 18 -2.8703 2.7141 2598 2743 1.53820 passages
10 19 -2.9165 2.7553 3487 3467 1.53040 passages
10 20 -2.7902 2.7916 4661 4603 1.52320 passages
10 21 -2.7654 2.8220 5910 5777 1.51640 passages
10 22 -2.8169 2.9796 7322 7359 1.51000 passages
10 23 -2.8178 2.7989 8825 8898 1.50400 passages
10 24 -2.9076 2.8799 10602 10598 1.49832 passages
10 13 4.0% 5.90E-10 5.87E-070 passages
10 14 6.0% 6.23E-10 1.43E-060 passages
10 15 6.0% 7.81E-10 1.08E-060 passages
10 16 4.5% 5.32E-10 7.75E-070 passages
10 17 8.0% 5.85E-10 9.22E-070 passages
10 18 7.5% 6.59E-10 1.88E-060 passages
10 19 11.0% 5.15E-10 3.07E-060 passages
10 20 12.5% 3.93E-10 7.00E-070 passages
10 21 31.5% 5.64E-10 3.54E-060 passages
10 22 50.0% 1.15E-09 2.39E-060 passages
10 23 50.0% 1.34E-09 3.11E-060 passages
10 24 50.0% 2.68E-09 6.82E-062 passages
31 passage
454 passages
01 passage
94 passages
91 passage
M2 passages
94 passages
324003 passages
10 13 0.0005330 4.127 0.1097 5.232 0.28 0.710 passages
10 14 0.0009764 4.236 0.1213 5.349 0.87 0.940 passages
10 15 0.0005171 4.154 0.1003 5.434 0.26 0.460 passages
10 16 0.0005202 4.202 0.1029 5.433 0.27 0.530 passages
10 17 0.0006583 4.183 0.0966 5.395 0.47 0.360 passages
10 18 0.0004390 4.194 0.1080 5.511 0.17 0.670 passages
10 19 0.0004969 4.200 0.0874 5.341 0.24 0.180 passages
10 20 0.0004351 4.268 0.1067 5.717 0.17 0.630 passages
10 21 0.0004934 4.316 0.1019 5.421 0.23 0.500 passages
10 22 0.0008161 4.347 0.1060 5.332 0.70 0.610 passages
10 23 0.0004249 4.304 0.1112 5.478 0.15 0.750 passages
10 24 0.0002799 4.158 0.0877 5.526 0.05 0.194 passages
N+1 � n � N+M9 passages
10 13 0.17608 0.03512 0.09608 0.05933 0.10107 0.1095 0.1719 0.24710 passages
10 14 0.17657 0.03540 0.09663 0.05990 0.10199 0.1108 0.1741 0.25100 passages
10 15 0.17697 0.03565 0.09710 0.06040 0.10277 0.1119 0.1759 0.25390 passages
10 16 0.17732 0.03586 0.09750 0.06084 0.10347 0.1129 0.1776 0.25670 passages
10 17 0.17760 0.03605 0.09785 0.06123 0.10407 0.1137 0.1789 0.25900 passages
10 18 0.17784 0.03621 0.09816 0.06157 0.10462 0.1145 0.1803 0.26130 passages
10 19 0.17805 0.03636 0.09843 0.06189 0.10511 0.1152 0.1814 0.26310 passages
10 20 0.17824 0.03649 0.09867 0.06215 0.10553 0.1158 0.1824 0.26490 passages
10 21 0.17839 0.03661 0.09888 0.06242 0.10595 0.1165 0.1836 0.26680 passages
10 22 0.17853 0.03671 0.09906 0.06262 0.10627 0.1169 0.1842 0.26780 passages
10 23 0.17864 0.03680 0.09922 0.06282 0.10658 0.1174 0.1850 0.26920 passages
10 24 0.17875 0.03688 0.09937 0.06301 0.10689 0.1179 0.1859 0.27080 passages
GUE 0.17999 0.03796 0.10130 0.06552 0.11096 0.1243 0.1969 0.29024 passages
10 13 0.23717 0.02671 0.16887 0.06252 0.2073 0.1530 0.3764 0.43040 passages
10 14 0.23846 0.02678 0.17045 0.06301 0.2099 0.1550 0.3827 0.43880 passages
10 15 0.23956 0.02688 0.17181 0.06349 0.2122 0.1568 0.3880 0.44580 passages
10 16 0.24050 0.02700 0.17299 0.06396 0.2142 0.1585 0.3927 0.45230 passages
10 17 0.24132 0.02713 0.17404 0.06446 0.2159 0.1601 0.3970 0.45830 passages
10 18 0.24202 0.02726 0.17494 0.06488 0.2175 0.1614 0.4005 0.46300 passages
10 19 0.24264 0.02740 0.17574 0.06530 0.2188 0.1627 0.4036 0.46720 passages
10 20 0.24319 0.02753 0.17645 0.06569 0.2201 0.1639 0.4065 0.47130 passages
10 21 0.24366 0.02766 0.17709 0.06609 0.2212 0.1651 0.4092 0.47530 passages
10 22 0.24409 0.02778 0.17765 0.06643 0.2222 0.1660 0.4114 0.47800 passages
10 23 0.24447 0.02790 0.17819 0.06679 0.2232 0.1671 0.4140 0.48210 passages
10 24 0.24480 0.02801 0.17863 0.06709 0.2240 0.1679 0.4158 0.48460 passages
GUE 0.249 0.03 0.185 0.073 0.237 0.185 0.451 0.5441 passage
Height log d n 1/ d n 1/ d n 20 passages
10 13 -0.101540 1.27050 2.526880 passages
10 14 -0.101798 1.27124 2.531730 passages
10 15 -0.102009 1.27184 2.540680 passages
10 16 -0.102188 1.27235 2.540680 passages
10 17 -0.102329 1.27272 2.540490 passages
10 18 -0.102453 1.27308 2.545400 passages
10 19 -0.102558 1.27338 2.549060 passages
10 20 -0.102650 1.27363 2.549960 passages
10 21 -0.102721 1.27382 2.549900 passages
10 22 -0.102789 1.27401 2.547830 passages
10 23 -0.102843 1.27415 2.551660 passages
10 24 -0.102891 1.27427 2.557280 passages
GUE -0.1035 1.2758 2.563313 passages
References52 references
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