Prime-Counting Function/Approximations

Approximations to Prime-Counting Function

A table of some of the values of the prime-counting ($\pi$) function compared with $\dfrac x {\ln x}$ and the Eulerian logarithmic integral $\ds \map \Li x = \int_2^x \frac {\d t} {\map \ln t}$:

$n$	$\map \pi n$	$\dfrac x {\ln x}$	$\map \Li x$
$1 \, 000$	$168$	$145$	$178$
$10 \, 000$	$1 \, 229$	$1 \, 068$	$1 \, 246$
$100 \, 000$	$9 \, 596$	$8 \, 686$	$9 \, 630$
$1 \, 000 \, 000$	$78 \, 498$	$72 \, 382$	$78 \, 628$
$10 \, 000 \, 000$	$664 \, 579$	$620 \, 421$	$664 \, 918$

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes