Definition:Prime-Counting Function/Approximations

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Definition

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

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


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