# 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.

$n$ 2 3 $\pi(n)$ $\int_{2}^{n}\frac{dt}{\log t}$ $\frac{n}{\log n}$ 1 1 3 2 2 3 3 4 3 4 5 4 4 6 4 5 7 5 9 11 7 10 13 9 24 28 20 25 29 21 25 30 22 29 32 23 30 33 24 31 36 26 99 105 84 100 108 86 145 153 123 154 161 131 155 164 133 168 178 145 185 193 158 189 196 161 190 199 163 217 224 185 218 229 189 237 246 204 289 299 249 1183 1197 1042 1184 1201 1046 1229 1246 1086 1831 1847 1623 1832 1852 1628 2225 2249 1984 2226 2254 1989 2652 2672 2365 3385 3412 3032 3386 3419 3038 9520 9555 8618 9592 9630 8686 78498 78628 72382 664579 664918 620421 5761455 5762209 5428681 50847534 50849235 48254942 455052511 455055615 434294482

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

Title values of the prime counting function and estimates for selected inputs ValuesOfThePrimeCountingFunctionAndEstimatesForSelectedInputs 2013-03-22 16:38:53 2013-03-22 16:38:53 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Example msc 11A41