WEBMar 17, 2020 · The de la Vallée-Poussin theorem on the distribution of prime numbers: Let $\pi (x)$ be the number of primes smaller than $x$; then, if $x \ge 1$, the following equality is valid: $$ \pi (x) = \mathrm {li} (x) + O\left ( { x \exp (-C\sqrt {\log x}) }\right) $$ where $C$ is a positive constant and $\mathrm {li} (x)$ is the logarithmic integral ...