Michael Rubinstein (Waterloo)
Lower terms in the moments of $L$-functions
A 100 year old problem asks to determine the moments of the Riemann zeta
function on Re(s)=1/2. The second and fourth moments are
well understood, but little has been proven about the higher moments.
These moments are needed to understand the distribution of the zeta function
and its extreme behaviour.
In recent years, a detailed conjectural picture has emerged concerning the full
asymptotics of the moments of the zeta function and other L-functions. I will
describe these developments and will also discuss the case of elliptic curve
L-functions, where knowledge of the moments has been used successfully to
make detailed predictions for ranks.