The Prime Number Theorem and Generalizations

Abstract

It was a century ago that Jacques Hadamard and Charles de la Vallée Poussin proved (independently) the celebrated prime number theorem. If π(x) denotes the number of primes up to x, the theorem states that

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