Moments of the Riemann zeta function

The typical behaviour of the values of the Riemann zeta function $\zeta(s)$ when the real part of $s$ is $1/2$
is typically studied via the moments $$\int_{0}^T  |\zeta(\frac{1}{2}+it )|^{2k} \mathrm{d}t,
(k \in \mathbb{N}).$$

 

The growth of these moments are conjecturally predicted via random matrix theory.

In this talk we shall briefly explain this connection and focus on the (conditional under RH)

proof of the conjectured growth by Soundararajan [Ann. of Math., 2009].