Adam J. Harper, Cambridge University, Cambridge, Royaume Uni

Sharp bounds for moments of the Riemann zeta function

The Riemann zeta function ζ(s) has been studied for more than 150 years, but our knowledge
about it remains very incomplete. On or near the critical line Re(s)=1/2, our knowledge is
lacking even if we assume the truth of the Riemann Hypothesis. For example, the behaviour
of the power moments ∫₀^T |ζ(1/2+it)|^2k dt, which is subject to precise conjectures coming from
random matrix theory, has resisted most rigorous study until recently.

In this talk I will try to explain work of Soundararajan, which gave nearly sharp upper bounds
for the moments of zeta (assuming the Riemann Hypothesis), and also my recent improvement
giving sharp upper bounds (assuming the Riemann Hypothesis).