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 $x \rightarrow \infty$ , 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: \begin{equation*} \pi(x)=\li x+R(x), \end{equation*}where $\li$ is the logarithmic integral defined as

for any fixed

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 Erdos and Selberg [3,6] found such a proof. Nowadays there are some very short proofs of the prime number theorem (for example, see [5]).

Bibliography

1

Tom M. Apostol.
Introduction to Analytic Number Theory.
Narosa Publishing House, second edition, 1990.
Zbl 0335.10001.

2

Harold Davenport.
Multiplicative Number Theory.
Markham Pub. Co., 1967.
Zbl 0159.06303.

3

Paul Erdos.
On a new method in elementary number theory.
Proc. Nat. Acad. Sci. U.S.A., 35:374-384, 1949.
Zbl 0034.31403.

4

Jacques Hadamard.
Sur la distribution des zéros de la fonction $\zeta(s)$ et ses conséquences arithmétiques.
Bull. Soc. Math. France, 24:199-220.
JFM 27.0154.01.

5

Donald J. Newman.
Simple analytic proof of the prime number theorem.
Amer. Math. Monthly, 87:693-696, 1980.
Available online at JSTOR.

6

Atle Selberg.
An elementary proof of the prime number theorem.
Ann. Math. (2), 50:305-311, 1949.
Zbl 0036.30604.

7

Charles de la Vallée Poussin.
Recherces analytiques sur la théorie des nombres premiers.
Ann. Soc. Sci. Bruxells, 1897.