Prime-counting Function - Inequalities

Inequalities

Here are some useful inequalities for π(x).

\frac {x} {\log x} < \pi(x) < 1.25506 \frac {x} {\log x} \! for x ≥ 17.

The left inequality holds for x ≥ 17 and the right inequality holds for x > 1.

An explanation of the constant 1.25506 is given at (sequence A209883 in OEIS).

\frac {x} {\log x + 2} < \pi(x) < \frac {x} {\log x - 4} \! for x ≥ 55.

Here are some inequalities for the nth prime, pn.

n\ \ln n + n\ln\ln n - n < p_n < n \ln n + n \ln \ln n \! for n ≥ 6.

The left inequality holds for n ≥ 1 and the right inequality holds for n ≥ 6.

An approximation for the nth prime number is

p_n = n \ln n + n \ln \ln n - n + \frac {n \ln \ln n - 2n} {\ln n} + O\left( \frac {n (\ln \ln n)^2} {(\ln n)^2}\right).

Read more about this topic:  Prime-counting Function

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