Prime number theorem

Statement

Statement in terms of the asymptotic distribution in ratio terms

This states that:

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

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:

$\left|\pi (x)-\operatorname {Li} (x)\right|=O(xe^{-a{\sqrt {\log x}}})$.

Here, $\operatorname {Li} (x)$ denotes the logarithmic integral.