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# values of the prime counting function and estimates for selected inputs

In illustrating the degree of error of various estimates of the prime counting function given in connection to the prime number theorem, it is customary to select powers of 10 as the inputs. These are given in the table below, mixed in with primes beginning and ending prime gaps, the fourth primes of selected prime quadruplets and selected record lows of the Mertens function. The values of the logarithmic integral and the division of $n$ by its natural logarithm have been rounded off to the nearest integer.

$n$ | $\pi(n)$ | $\int_{2}^{n}\frac{dt}{\log t}$ | $\frac{n}{\log n}$ |
---|---|---|---|

2 | 1 | 1 | 3 |

3 | 2 | 2 | 3 |

5 | 3 | 4 | 3 |

7 | 4 | 5 | 4 |

10 | 4 | 6 | 4 |

11 | 5 | 7 | 5 |

23 | 9 | 11 | 7 |

29 | 10 | 13 | 9 |

89 | 24 | 28 | 20 |

97 | 25 | 29 | 21 |

100 | 25 | 30 | 22 |

110 | 29 | 32 | 23 |

113 | 30 | 33 | 24 |

127 | 31 | 36 | 26 |

523 | 99 | 105 | 84 |

541 | 100 | 108 | 86 |

829 | 145 | 153 | 123 |

887 | 154 | 161 | 131 |

907 | 155 | 164 | 133 |

1000 | 168 | 178 | 145 |

1105 | 185 | 193 | 158 |

1129 | 189 | 196 | 161 |

1151 | 190 | 199 | 163 |

1327 | 217 | 224 | 185 |

1361 | 218 | 229 | 189 |

1489 | 237 | 246 | 204 |

1879 | 289 | 299 | 249 |

9551 | 1183 | 1197 | 1042 |

9587 | 1184 | 1201 | 1046 |

10000 | 1229 | 1246 | 1086 |

15683 | 1831 | 1847 | 1623 |

15727 | 1832 | 1852 | 1628 |

19609 | 2225 | 2249 | 1984 |

19661 | 2226 | 2254 | 1989 |

23833 | 2652 | 2672 | 2365 |

31397 | 3385 | 3412 | 3032 |

31469 | 3386 | 3419 | 3038 |

99139 | 9520 | 9555 | 8618 |

100000 | 9592 | 9630 | 8686 |

1000000 | 78498 | 78628 | 72382 |

10000000 | 664579 | 664918 | 620421 |

100000000 | 5761455 | 5762209 | 5428681 |

1000000000 | 50847534 | 50849235 | 48254942 |

10000000000 | 455052511 | 455055615 | 434294482 |

The smaller values (up to $n=2000$) have been verified by hand. Above that, I have trusted Mathematica 4.2 completely.

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