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 number theorem:

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

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 fully known. From the work of Korobov and Vinogradov on zeroes of Riemann 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 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 $\mathrm{\Re }s=1$ 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]).

Title	prime number theorem
Canonical name	PrimeNumberTheorem
Date of creation	2013-03-22 11:45:18
Last modified on	2013-03-22 11:45:18
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	21
Author	bbukh (348)
Entry type	Theorem
Classification	msc 11A41
Classification	msc 55U10
Classification	msc 55U30
Classification	msc 55T25
Classification	msc 55M05
Classification	msc 55U15
Classification	msc 81T25
Classification	msc 81-XX
Classification	msc 20G42
Classification	msc 81R50
Classification	msc 17B37
Classification	msc 81Q60
Classification	msc 81V05
Classification	msc 81T05
Classification	msc 55R40
Defines	pi(x)
Defines	logarithmic integral