Maxim Gerspach: Central Limit Theorems in Number Theory

In this talk I will survey some of the starting points of probabilistic number theory, focusing on two central limit theorems for arithmetic quantities. The first one is the Erdős–Kac Theorem, which states that the number of prime factor of a randomly chosen number of size x tends to a normal distribution with mean and variance log log x. The second is known as the Selberg central limit theorem and states that the logarithm of the Riemann zeta function evaluated at a random point on the critical line with real part 1/2, suitably normalised, converges in law to a complex Gaussian random variable.

This talk precedes Adam Harper's colloquium talk  on “Multiplicative functions and probability” and is accessible to a broad audience (master and PhD students).