The prime number theorem and Selberg’s method

Abstract

Let π(x) denote, for any real x, the number of primes not exceeding x. The prime number theorem is the assertion that

$$ \mathop{{\lim}}\limits_{{x\to\infty}}(\frac{{\pi (x)}}{{x/\log x}})=1 $$

((1))

A fundamental formula discovered by Atle Selberg has made a proof of (1) possible without the use of the properties of the zeta-function of Riemann, and without the use of the theory of functions of a complex variable. We shall prove Selberg’s formula in this chapter, and indicate some of its consequences. We shall also prove an inequality due to E. Wirsing, which, when combined with a variant of Selberg’s formula, gives a proof of the prime number theorem.