The square of the Riemann zeta function gives rise to critical multiplicative chaos

Multiplicative chaos is a class of random measures, that have recently been found to have strong connections with number
theoretic objects like L-functions and character sums, and the phenomenon of ''better than squareroot cancellation''.
Saksman and Webb have conjectured that integrating test functions against absolute powers of the Riemann zeta function
should give rise to these measures. The square of the zeta function is particularly interesting, since this should correspond
to the so-called critical chaos. I will report on joint work (in preparation) of myself, Saksman and Webb, which proves
their conjecture for zeta squared. I will try to give a gentle introduction to these problems, and also indicate some of the
main proof ideas, which may be of independent interest.