Prime Number Theorem - Bounds On The Prime-counting Function

Bounds On The Prime-counting Function

The prime number theorem is an asymptotic result. Hence, it cannot be used to bound π(x).

However, some bounds on π(x) are known, for instance Pierre Dusart's

The first inequality holds for all x ≥ 599 and the second one for x ≥ 355991.

A weaker but sometimes useful bound is

for x ≥ 55. In Dusart's thesis there are stronger versions of this type of inequality that are valid for larger x.

The proof by de la Vallée-Poussin implies the following. For every ε > 0, there is an S such that for all x > S,