Define $\pi(x)$ as the number of primes less than or equal to $x$. The prime number theorem asserts that

as $x\rightarrow\infty$, that is, $\pi(x)/\frac{x}{\log x}$ tends to 1 as $x$ increases. Here ${\log x}$ is the natural logarithm.

There is a sharper statement that is also known as the prime number theorem:

where $\li$ is the logarithmic integral defined as

and $R(x)$ is the error term whose behavior is still not fully known. From the work of Korobov and Vinogradov on zeroes of Riemann zeta-function it is known that

for every $\theta>\tfrac{3}{5}$. The unproven Riemann hypothesis is equivalent to the statement that $R(x)=O(x^{{1/2}}\log x)$.

There exist a number of proofs of the prime number theorem. The original proofs by Hadamard [4] and de la Vallée Poussin[7] called on analysis of behavior of the Riemann zeta function $\zeta(s)$ near the line $\Re s=1$ to deduce the estimates for $R(x)$. For a long time it was an open problem to find an elementary proof of the prime number theorem (“elementary” meaning “not involving complex analysis”). Finally Erdős and Selberg [3, 6] found such a proof. Nowadays there are some very short proofs of the prime number theorem (for example, see [5]).