Definition:Prime-Counting Function/Approximations

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 $\displaystyle \operatorname {Li} \left({x}\right) = \int_2^x \frac {\mathrm d t} {\ln \left({t}\right)}$:


 * {| border="1"

! align="right" style = "padding: 2px 10px" | $n$ ! align="right" style = "padding: 2px 10px" | $\pi \left({n}\right)$ ! align="right" style = "padding: 2px 10px" | $\dfrac x {\ln x}$ ! align="right" style = "padding: 2px 10px" | $\operatorname {Li} \left({x}\right)$
 * align="right" style = "padding: 2px 10px" | $1 \, 000$
 * align="right" style = "padding: 2px 10px" | $168$
 * align="right" style = "padding: 2px 10px" | $145$
 * align="right" style = "padding: 2px 10px" | $178$
 * align="right" style = "padding: 2px 10px" | $10 \, 000$
 * align="right" style = "padding: 2px 10px" | $1 \, 229$
 * align="right" style = "padding: 2px 10px" | $1 \, 068$
 * align="right" style = "padding: 2px 10px" | $1 \, 246$
 * align="right" style = "padding: 2px 10px" | $100 \, 000$
 * align="right" style = "padding: 2px 10px" | $9 \, 596$
 * align="right" style = "padding: 2px 10px" | $8 \, 686$
 * align="right" style = "padding: 2px 10px" | $9 \, 630$
 * align="right" style = "padding: 2px 10px" | $1 \, 000 \, 000$
 * align="right" style = "padding: 2px 10px" | $78 \, 498$
 * align="right" style = "padding: 2px 10px" | $72 \, 382$
 * align="right" style = "padding: 2px 10px" | $78 \, 628$
 * align="right" style = "padding: 2px 10px" | $10 \, 000 \, 000$
 * align="right" style = "padding: 2px 10px" | $664 \, 579$
 * align="right" style = "padding: 2px 10px" | $620 \, 421$
 * align="right" style = "padding: 2px 10px" | $664 \, 918$
 * }
 * align="right" style = "padding: 2px 10px" | $10 \, 000 \, 000$
 * align="right" style = "padding: 2px 10px" | $664 \, 579$
 * align="right" style = "padding: 2px 10px" | $620 \, 421$
 * align="right" style = "padding: 2px 10px" | $664 \, 918$
 * }