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[parent] values of the prime counting function and estimates for selected inputs (Example)

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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Cross-references: Mathematica, integer, natural logarithm, division, logarithmic integral, Mertens function, prime quadruplets, prime gaps, primes, prime number theorem, connection, prime counting function, estimates, degree

This is version 1 of values of the prime counting function and estimates for selected inputs, born on 2007-01-31.
Object id is 8852, canonical name is ValuesOfThePrimeCountingFunctionAndEstimatesForSelectedInputs.
Accessed 454 times total.

Classification:
AMS MSC: 11A41 (Number theory :: Elementary number theory :: Primes)
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