Table of π(x), x / Ln x, and Li(x)
The table compares exact values of π(x) to the two approximations x / ln x and li(x). The last column, x / π(x), is the average prime gap below x.
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x π(x) π(x) − x / ln x π(x) / (x / ln x) li(x) − π(x) x / π(x) 10 4 −0.3 0.921 2.2 2.500 102 25 3.3 1.151 5.1 4.000 103 168 23 1.161 10 5.952 104 1,229 143 1.132 17 8.137 105 9,592 906 1.104 38 10.425 106 78,498 6,116 1.084 130 12.740 107 664,579 44,158 1.071 339 15.047 108 5,761,455 332,774 1.061 754 17.357 109 50,847,534 2,592,592 1.054 1,701 19.667 1010 455,052,511 20,758,029 1.048 3,104 21.975 1011 4,118,054,813 169,923,159 1.043 11,588 24.283 1012 37,607,912,018 1,416,705,193 1.039 38,263 26.590 1013 346,065,536,839 11,992,858,452 1.034 108,971 28.896 1014 3,204,941,750,802 102,838,308,636 1.033 314,890 31.202 1015 29,844,570,422,669 891,604,962,452 1.031 1,052,619 33.507 1016 279,238,341,033,925 7,804,289,844,393 1.029 3,214,632 35.812 1017 2,623,557,157,654,233 68,883,734,693,281 1.027 7,956,589 38.116 1018 24,739,954,287,740,860 612,483,070,893,536 1.025 21,949,555 40.420 1019 234,057,667,276,344,607 5,481,624,169,369,960 1.024 99,877,775 42.725 1020 2,220,819,602,560,918,840 49,347,193,044,659,701 1.023 222,744,644 45.028 1021 21,127,269,486,018,731,928 446,579,871,578,168,707 1.022 597,394,254 47.332 1022 201,467,286,689,315,906,290 4,060,704,006,019,620,994 1.021 1,932,355,208 49.636 1023 1,925,320,391,606,803,968,923 37,083,513,766,578,631,309 1.020 7,250,186,216 51.939 OEIS A006880 A057835 A057752
Read more about this topic: Prime Number Theorem
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