Different Approaches to the Distribution of Primes

In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zeta-function. 1. The prime number theorem, from the beginning By studying tables of primes, Gauss understood, as a boy of 15 or 16 (in 1792 or 1793), that the primes occur with density 1 log x at around x. In other words π(x) := #{primes ≤ x} ≈ Li(x) where Li(x) := ∫ x