Chebyshev’s Theorem on the Asymptotic Density of the Primes

Abstract

Let π(n) denote the number of primes that are no larger than n; that is,

$$\displaystyle{\pi (n) =\sum _{p\leq n}1,}$$

where here and elsewhere in this chapter and the next two, the letter p in a summation denotes a prime. Euclid proved that there are infinitely many primes: $\lim _{n\rightarrow \infty }\pi (n) = \infty$. The asymptotic density of the primes is 0; that is,

$$\displaystyle{ \lim _{n\rightarrow \infty }\frac{\pi (n)} {n} = 0. }$$

The prime number theorem gives the leading order asymptotic behavior of π(n).