Some Elementary Theorems on the Distribution of Prime Numbers

Abstract

If x > 0 let π(x) denote the number of primes not exceeding x. Then π(x) → ∞ as x → ∞ since there are infinitely many primes. The behavior of π(x) as a function of x has been the object of intense study by many celebrated mathematicians ever since the eighteenth century. Inspection of tables of primes led Gauss (1792) and Legendre (1798) to conjecture that π(x) is asymptotic to x/log x, that is,

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