素数定理
定理の内容 > π(x), x/log(x), li(x) の表

表は π(x) 、 x/log(x) 、 li(x) の値とそれらの比較の表である。
| x | π(x) | π(x) − x/log(x) | π(x)/(x/log(x)) | li(x) − π(x) | x/π(x) | x/log(x) | li(x) |
| --- | --- | --- | --- | --- | --- | --- | --- |
| 10 | 4 | −0.343 | 0.921 | 2.166 | 2.500 | 4.343 | 6.166 |
| 102 | 25 | 3.285 | 1.151 | 5.126 | 4.000 | 21.715 | 30.126 |
| 103 | 168 | 23 | 1.161 | 10 | 5.952 | 145 | 178 |
| 104 | 1,229 | 143 | 1.132 | 17 | 8.137 | 1,086 | 1,246 |
| 105 | 9,592 | 906 | 1.104 | 38 | 10.425 | 8,686 | 9,630 |
| 106 | 78,498 | 6,116 | 1.084 | 130 | 12.740 | 72,382 | 78,628 |
| 107 | 664,579 | 44,158 | 1.071 | 339 | 15.047 | 620,421 | 664,918 |
| 108 | 5,761,455 | 332,774 | 1.061 | 754 | 17.357 | 5,428,681 | 5,762,209 |
| 109 | 50,847,534 | 2,592,592 | 1.054 | 1,701 | 19.667 | 48,254,942 | 50,849,235 |
| 1010 | 455,052,511 | 20,758,029 | 1.048 | 3,104 | 21.975 | 434,294,482 | 455,055,615 |
| 1011 | 4,118,054,813 | 169,923,159 | 1.043 | 11,588 | 24.283 | 3,948,131,654 | 4,118,066,401 |
| 1012 | 37,607,912,018 | 1,416,705,193 | 1.039 | 38,263 | 26.590 | 36,191,206,825 | 37,607,950,281 |
| 1013 | 346,065,536,839 | 11,992,858,452 | 1.034 | 108,971 | 28.896 | 334,072,678,387 | 346,065,645,810 |
| 1014 | 3,204,941,750,802 | 102,838,308,636 | 1.033 | 314,890 | 31.202 | 3,102,103,442,166 | 3,204,942,065,692 |
| 1015 | 29,844,570,422,669 | 891,604,962,452 | 1.031 | 1,052,619 | 33.507 | 28,952,965,460,217 | 29,844,571,475,288 |
| 1016 | 279,238,341,033,925 | 7,804,289,844,393 | 1.029 | 3,214,632 | 35.812 | 271,434,051,189,532 | 279,238,344,248,557 |
| 1017 | 2,623,557,157,654,233 | 68,883,734,693,281 | 1.027 | 7,956,589 | 38.116 | 2,554,673,422,960,305 | 2,623,557,165,610,822 |
| 1018 | 24,739,954,287,740,860 | 612,483,070,893,536 | 1.025 | 21,949,555 | 40.420 | 24,127,471,216,847,324 | 24,739,954,309,690,415 |
| 1019 | 234,057,667,276,344,607 | 5,481,624,169,369,960 | 1.024 | 99,877,775 | 42.725 | 228,576,043,106,974,646 | 234,057,667,376,222,382 |
| 1020 | 2,220,819,602,560,918,840 | 49,347,193,044,659,701 | 1.023 | 222,744,644 | 45.028 | 2,171,472,409,516,259,138 | 2,220,819,602,783,663,484 |
| 1021 | 21,127,269,486,018,731,928 | 446,579,871,578,168,707 | 1.022 | 597,394,254 | 47.332 | 20,680,689,614,440,563,221 | 21,127,269,486,616,126,182 |
| 1022 | 201,467,286,689,315,906,290 | 4,060,704,006,019,620,994 | 1.021 | 1,932,355,208 | 49.636 | 197,406,582,683,296,285,296 | 201,467,286,691,248,261,498 |
| 1023 | 1,925,320,391,606,803,968,923 | 37,083,513,766,578,631,309 | 1.020 | 7,250,186,216 | 51.939 | 1,888,236,877,840,225,337,614 | 1,925,320,391,614,054,155,139 |
| 1024 | 18,435,599,767,349,200,867,866 | 339,996,354,713,708,049,069 | 1.019 | 17,146,907,278 | 54.243 | 18,095,603,412,635,492,818,797 | 18,435,599,767,366,347,775,144 |
| 1025 | 176,846,309,399,143,769,411,680 | 3,128,516,637,843,038,351,228 | 1.018 | 55,160,980,939 | 56.546 | 173,717,792,761,300,731,060,452 | 176,846,309,399,198,930,392,619 |
| 1026 | 1,699,246,750,872,437,141,327,603 | 28,883,358,936,853,188,823,261 | 1.017 | 155,891,678,121 | 58.850 | 1,670,363,391,935,583,952,504,342 | 1,699,246,750,872,593,033,005,724 |
| 1027 | 16,352,460,426,841,680,446,427,399 | 267,479,615,610,131,274,163,365 | 1.0166 | 508,666,658,006 | 61.153 | 16,084,980,811,231,549,172,264,034 | 16,352,460,426,842,189,113,085,405 |
| 1028 | 157,589,269,275,973,410,412,739,598 | 2,484,097,167,669,186,251,622,127 | 1.016 | 1,427,745,660,374 | 63.456 | 155,105,172,108,304,224,161,117,471 | 157,589,269,275,974,838,158,399,972 |
π(1024) の値はもともとリーマン予想を仮定して計算されたが、その後無条件で証明された。