# 素數定理

（重定向自素數定理

${\displaystyle \pi (x)\approx {\frac {x}{\ln \,x}}}$

${\displaystyle \pi (x)={\rm {Li}}(x)+O\left(xe^{-{\frac {1}{15}}{\sqrt {\ln \,x}}}\right)}$，當x 趨近∞。

## 敘述

${\displaystyle \lim _{x\to \infty }{\frac {\;\pi (x)\;}{\frac {x}{\ln(x)}}}=1}$

${\displaystyle \pi (x)\sim {\frac {x}{\ln x}}}$

## 關於 π(x)、x / ln x 和 li(x) 的數值

${\displaystyle x}$ ${\displaystyle {\boldsymbol {\pi }}(x)}$[1] ${\displaystyle {\boldsymbol {\pi }}(x)-{\frac {x}{\ln x}}}$[2] ${\displaystyle {\frac {{\boldsymbol {\pi }}(x)}{\frac {x}{\ln x}}}}$ ${\displaystyle {\rm {Li}}(x)-{\boldsymbol {\pi }}(x)}$[3] ${\displaystyle {\frac {x}{{\boldsymbol {\pi }}(x)}}}$
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
1024 18,435,599,767,349,200,867,866 339,996,354,713,708,049,069 1.019 17,146,907,278 54.243
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
OEIS A006880 A057835 A057752

${\displaystyle p(n)\sim n\ln \,n.}$

## 歷史

${\displaystyle \pi (x)={\rm {Li}}(x)+O\left({\sqrt {x}}\ln \,x\right)}$