prime number theorem

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

$$\pi (x)\sim \frac{x}{\log x}$$

as $x\to \mathrm{\infty }$, 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 numbertheorem:

$$\pi (x)=\mathrm{li}x+R(x),$$

where $\mathrm{li}$ is the logarithmic integral defined as

$$\mathrm{li}x={\int }_2^x\frac{dt}{\log t}=\frac{x}{\log x}+\frac{1!x}{{(\log x)}^2}+\mathrm{\cdots }+\frac{(k-1)!x}{{(\log x)}^k}+O\left\{\frac{x}{{(\log x)}^{k+1}}\right\},\text{for any fixed }k$$

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

$$R(x)=O\left\{x\exp (-c(\theta ){(\log x)}^{\theta })\right\}$$

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

There exist a number of proofs of the prime number theorem. Theoriginal proofs by Hadamard [4] and de laVallée Poussin[7] called on analysis ofbehavior of the Riemann zeta function $\zeta (s)$ near the line $\mathrm{\Re }s=1$to deduce the estimates for $R(x)$. For a long time it was an openproblem 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 numbertheorem (for example, see [5]).