structural:expansion, linguistic:voice_change
Inequalities

Here are some useful inequalities for π (x).

for x ≥ 17.

The left inequality holds for x ≥ 17 and the right inequality holds for x > 1. The constant 1.25506 is 30 log 113 / 113 to 5 decimal places, as π (x) log x / x has its maximum value at x = 113.

Pierre Dusart proved in 2010:

and

Here are some inequalities for the n th prime, p. The upper bound is due to Rosser (1941), the lower one to Dusart (1999):

n (log (n log n) − 1) < p n < n log (n log n) for n ≥ 6. {\displaystyle n(\log(n\log n)-1)<p_{n}<n{\log(n\log n)}\quad {\text{for }}n\geq 6.}

The left inequality holds for n ≥ 2 and the right inequality holds for n ≥ 6.

An approximation for the n th prime number is

Ramanujan proved that the inequality

holds for all sufficiently large values of x.

In 2010 Dusart proved (Proposition 6.6) that, for n ≥ 688383,

and (Proposition 6.7) that, for n ≥ 3,

More recently, Dusart has proved (Theorem 5.1) that, for x > 1,

and that, for x ≥ 88789,