素数分布数据
x p(x) Li(x)-p(x) q(x) Comments 106 78,498 129 -0.209107 664,579 336 -0.1809
108 5,761,455 753 -0.1339 Meissel 1871 109 50,847,534 1,700 -0.0994 Meissel 1886 (Corrected)1010 455,052,511 3,103 -0.0594 Lehmer 1959 (Corrected) 1011 4,118,054,813 11,587 -0.0725
1012 37,607,912,018 38,262 -0.0781
1013 346,065,536,839 108,970 -0.0723 Bohmann 1972 (Corrected) 1014 3,204,941,750,802 314,889 -0.0678 Lagarias Miller Odlyzko 1985 1015 29,844,570,422,669 1,052,618 -0.0735 LMO 1985 1016 279,238,341,033,925 3,214,631 -0.0726 LMO 1985 2*1016 547,863,431,950,008 3,776,488 -0.0607 LMO 1985 4*1016 1,075,292,778,753,150 5,538,861 -0.0634 LMO 1985 (World record at that time, in 28 hours) 1017 2,623,557,157,654,233 7,956,588 -0.0581 Deléglise Rivat 1994 1.5*1017 3,893,882,469,583,623
1018 24,739,954,287,740,860 21,949,554 -0.05177 Deléglise Rivat 1994 2*1018 48,645,161,281,738,535 29,811,771 -0.0500 Deléglise 1996 3*1018 72,254,704,797,687,083 45,770,751 -0.0629 Deléglise 1996 4*1018 95,676,260,903,887,607 69,277,091 -0.0827 Deléglise 1996 5*1018 118,959,989,688,273,472 43,092,600 -0.0461 Deléglise 1996 6*1018 142,135,049,412,622,144 58,896,042 -0.0576 Deléglise 1996 7*1018 165,220,513,980,969,424 61,621,341 -0.0559 Deléglise 1996 8*1018 188,229,829,247,429,504 59,219,350 -0.0503 Deléglise 1996 9*1018 211,172,979,243,258,278 87,883,179 -0.0704 Deléglise 1996 1019 234,057,667,276,344,607 99,877,774 -0.0760 Deléglise 1996 2*1019 460,637,655,126,005,490 108,526,618 -0.0587 Deléglise 1996 3*1019 684,559,920,583,084,690 146,461,969 -0.0649 P. Demichel, X. Gourdon 2000 (1) 3.1*1019 706,849,401,776,940,275 146,921,468 -0.0641 P. Demichel, X. Gourdon 2000 (1) 4*1019 906,790,515,105,576,571 98,572,744 -0.0379 Deléglise 1996 5*1019 1,127,779,923,790,184,543 133,700,534 -0.0461 P. Demichel, X. Gourdon 2000 (1) 1020 2,220,819,602,560,918,840 222,744,643 -0.0546 Deléglise 1996 (World record at that time, in 11 days) 1.6*1020 3,516,585,752,930,430,595 278,427,046 -0.0542P. Demichel, X. Gourdon 2001 Feb (1)1.7*1020 3,731,397,829,842,461,168206,233,344 -0.0389P. Demichel, X. Gourdon 2001 Feb (1)1.8*1020 3,945,938,652,811,699,917237,941,373 -0.0437P. Demichel, X. Gourdon 2001 Feb (1)2*1020 4,374,267,703,076,959,271 472,270,046 -0.0823 X. Gourdon 2000Nov 2.1*1020 4,588,082,544,160,859,769336,320,666 -0.0572P. Demichel, X. Gourdon 2001 Feb (1)2.6*1020 5,654,086,442,526,321,042485,136,141 -0.0743P. Demichel, X. Gourdon 2001 Feb (1)2.7*1020 5,866,733,862,193,360,875283,701,595 -0.0427P. Demichel, X. Gourdon 2001 Feb (1)3*1020 6,503,696,293,016,202,398 479,876,029 -0.0686 P. Demichel, X. Gourdon 2000 Dec (1)3.1*1020 6,715,709,842,660,010,419 359,287,610 -0.0505P. Demichel, X. Gourdon 2001 Feb (1)3.2*1020 6,927,578,466,326,617,308475,518,185 -0.0658P. Demichel, X. Gourdon 2001 Feb (1)4*1020 8,617,821,096,373,621,600 272,942,796 -0.0338 P. Demichel, X. Gourdon 2000 Dec (1)5*1020 10,720,710,117,789,005,897 387,506,247 -0.0430 P. Demichel, X. Gourdon 2000 Dec (1)6*1020 12,814,731,195,053,369,962 ??? ???P. Demichel, X. Gourdon 2001 Jan (1)1021 21,127,269,486,018,731,928 597,394,253 -0.0471 X. Gourdon 2000Nov 2*1021 41,644,391,885,053,857,293 1,454,564,714 -0.0816 pi(x) project, 2000 Dec 4*1021 82,103,246,362,658,124,007 1,200,472,717 -0.0479 pi(x) project, 2000 Dec 1022 201,467,286,689,315,906,290 1,932,355,207 -0.0491 pi(x) project, 2000 Dec 1.5*1022 299,751,248,358,699,805,270 2,848,114,312 -0.0592 P. Demichel, X. Gourdon, 2001 Feb (70 days on PIII 1 Ghz) (1)2*1022 397,382,840,070,993,192,736 2,732,289,619 -0.0493 pi(x) project, 2001 Feb 4*1022 783,964,159,847,056,303,858 5,101,648,384 -0.0655 pi(x) project, 2001 Mar (Current world record)
转自：http://numbers.computation.free.fr/Constants/Primes/pixtable.html
http://en.wikipedia.org/wiki/Prime-counting_function
http://primes.utm.edu/howmany.shtml 网页复制张贴后变样，10后面的乘方符号不见，请到实际网页查看 素数个数： 今天又看见一些新网站
http://oeis.org/A007053
pi(2^n),n=1..53之间的素数分布
http://www.primenumbers.net/Henri/us/ParPius.htm
这个是大的2的方幂素数分布数据
http://www.math.uni-bonn.de/people/jbuethe/topics/AnalyticPiX.html
涉及pi(2^77),pi(2^78),pi(2^79),pi(2^80),10^10～10^13之间每10亿为一个单位的素数分布
http://www.math.uni-bonn.de/people/jbuethe/pix-tables/pik1e10.k=10000-160000
http://www.primefan.ru/stuff/primes/table.html
网页的中间部分,有一个素数分布表,很多数据你想不到!共有20多个txt文本文件,喜欢收集的赶紧收集一下,可以进行一些异想天开数论猜想验证. 云梦 发表于 2013-4-12 17:05
素数个数：
忽然看到这个，正好之前看到一些好东西，你可以看看的
http://www.changhai.org/articles/science/mathematics/riemann_hypothesis/02.php
这里有一个有趣的细节值得一提： 从素数分布与素数定理的图示以及从大范围的计算中人们都发现 Li(x)-π(x) 大于零， 这使得有人猜测 Li(x) 不仅是素数分布的渐近形式， 而且还是其严格上界， 即 Li(x)-π(x) 恒大于零。 这种猜测在 1914 年被英国数学家 John Littlewood (1885-1977) 所推翻， Littlewood 证明了 Li(x)-π(x) 是一个在正与负之间振荡无穷多次的函数。 本帖最后由 熊一兵广义概率 于 2020-2-27 21:39 编辑
http://primes.utm.edu/howmany.shtml
1 10 4
2 100 25
3 1,000 168
4 10,000 1,229
5 100,000 9,592
6 1,000,000 78,498
7 10,000,000 664,579
8 100,000,000 5,761,455
9 1,000,000,000 50,847,534
10 10,000,000,000 455,052,511
11 100,000,000,000 4,118,054,813
12 1,000,000,000,000 37,607,912,018
13 10,000,000,000,000 346,065,536,839
14 100,000,000,000,000 3,204,941,750,802
15 1,000,000,000,000,000 29,844,570,422,669
16 10,000,000,000,000,000 279,238,341,033,925
17 100,000,000,000,000,000 2,623,557,157,654,233
18 1,000,000,000,000,000,000 24,739,954,287,740,860
19 10,000,000,000,000,000,000 234,057,667,276,344,607
20 100,000,000,000,000,000,000 2,220,819,602,560,918,840
21 1,000,000,000,000,000,000,000 21,127,269,486,018,731,928
22 10,000,000,000,000,000,000,000 201,467,286,689,315,906,290
23 100,000,000,000,000,000,000,000 1,925,320,391,606,803,968,923
24 1,000,000,000,000,000,000,000,000 18,435,599,767,349,200,867,866 (note)
25 10,000,000,000,000,000,000,000,000 176,846,309,399,143,769,411,680
网上早就有10的26次方的分布数据了，你怎么不一起公布 熊一兵广义概率 发表于 2020-3-30 19:24
熊一兵先生何时也来数学研发论坛了，我好久不来了，在看小说，今天有空浏览一下，不知先生是否有新的进展或对素数更新颖的体会。 这个玩意，可以用对数积分很好的估计，计算精确值很难的，
就好比判定一个整数是不是素数，用概率的办法真的很快，用确定性的办法真的很慢！
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