The talk traces early ideas about the distribution of primes, from Euler’s observations to the approximations of Legendre and Gauss for $\pi(x)$, the number of primes up to $x$. Euler’s product shows how the Riemann zeta function encodes information about primes. Riemann’s explicit formula reveals the role of its zeros in counting primes and their powers.

To explain the classical proof of the prime number theorem, the talk introduces the von Mangoldt function and Chebyshev’s function $\psi(x)$. The logarithmic derivative of the zeta function and residue calculus give an explicit formula for $\psi(x)$. The talk concludes with the role of de la Vallée Poussin’s zero-free region in proving that $\pi(x)\sim x/\log x$.