Prime number theorem

Statement

Statement in terms of the asymptotic distribution in ratio terms

This states that:

$\underset{x\to \mathrm{\infty }}{lim}\frac{\pi (x)}{x/\log x}=1$

where $\pi (x)$ is the prime-counting function: the number of primes less than or equal to $x$, and $\log x$ is the natural logarithm, i.e., the logarithm to base $e$.

Statement in terms of the logarithmic integral

This is a stronger formulation, which states that:

$|\pi (x)-\mathrm{L}\mathrm{i}(x)|=O(x{e}^{-a\sqrt{\log x}})$.

Here, $\mathrm{L}\mathrm{i}(x)$ denotes the logarithmic integral.