Georgia Institute of Technology College of Sciences

The moments of the Riemann zeta-function were introduced more than 100 years ago by Hardy and Littlewood, who showed that the Lindelof hypothesis (which provides a strong upper bound for the Riemann zeta-function on the critical line) is equivalent to obtaining sharp bounds on all the positive, even integral moments. Since then, the moments of the Riemann zeta-function and of more general L-functions have become natural objects of study. In this talk, I will review some of the history of the problem of evaluating moments, and focus on three different lines of research: studying negative moments of L-functions (which have been much less studied over the years, but which have rich applications nevertheless), computing lower-order terms in the moment asymptotics and obtaining non-vanishing results for L-functions evaluated at special points.