TheoremProved

Prime number theorem

Statement

Let $\pi(x)$ count the primes not exceeding $x$ . Then $\pi(x) \sim \dfrac{x}{\ln x}$ as $x\to\infty$ , i.e. $\displaystyle\lim_{x\to\infty}\frac{\pi(x)\ln x}{x} = 1$ .

Why is it true?

Near a large number $x$ , a randomly chosen integer has roughly a $1/\ln x$ chance of being prime — primes thin out logarithmically as numbers grow, not at any simpler rate such as polynomially.

Proof sketch

Connect $\pi(x)$ to the complex-analytic behaviour of the Riemann zeta function $\zeta(s)=\sum n^{-s}$ via its Euler product over primes. The key analytic input, established independently by Hadamard and de la Vallée Poussin, is that $\zeta(s)$ has no zeros on the line $\mathrm{Re}(s)=1$ . A Tauberian argument then converts this zero-free region into the asymptotic $\pi(x)\sim x/\ln x$ .

Stated by

stated 1792Carl Friedrich Gauss

Topics that use this theorem

Prime number theorem The Riemann zeta function and the Riemann hypothesis

Related theorems

Infinitude of primes (Euclid)Euler product formula for ζ(s)Dirichlet's theorem on primes in arithmetic progressions

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Jacques Hadamard (1896). Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques
Tom M. Apostol (1976). Introduction to Analytic Number Theory · DOI:10.1007/978-1-4757-5579-4