at the

American Institute of Mathematics, San Jose, California

organized by

Siegfred Baluyot, Steve Gonek, and Jon Keating

Despite their importance, rigorous proofs of asymptotic formulae for the moments are only known for a few cases. In the mid 90's, Conrey and Ghosh conjectured a formula for the sixth moment and later Conrey and Gonek developed a heuristic method based on moments of long Dirichlet polynomials that gave the second, fourth, sixth, and eighth moments. The method failed, however, for the tenth moment. At the same time, J. P. Keating and N. Snaith used techniques from random matrix theory to conjecture an asymptotic formula for all the moments. In the years that followed, this conjecture was made more precise and similar conjectures have been made for other families of $L$-functions by heuristic methods on the number theory side now known as ''the recipe'' and ''the ratios conjecture''.

The mystery behind the failure of the long Dirichlet polynomial method in the work of Conrey and Gonek has never been adequately understood. However, in a recent series of articles Conrey and Keating have revisited the issue and have given an in-depth analysis of the long polynomial approach that reveals why it fails after the eighth moment, and how it may be corrected. It is now emerging that there are neglected terms in this approach and that similar terms arise in a host of other problems such as in the variance of the divisor function in short intervals and in the variance of the divisor function in arithmetic progressions. It also turns out that the calculation of these terms is similar to that in the circle method.

The workshop has two main goals. The first is to apply the method to a wide range of other problems. This would give a new perspective to these problems that are otherwise currently intractable. The second goal is to begin work towards making the method rigorous. Among the specific objectives of the workshop are the following.

- To use the method to predict precise formulae for moments of $L$-functions in various families such as quadratic Dirichlet $L$-functions and cusp form $L$-functions
- To work out analogues for $L$-functions in the function field setting
- To carry out rigorous proofs for low moments in various families of $L$-functions
- To apply the method to divisor sums and other arithmetical questions
- To explore connections with the circle method

The workshop schedule.

A report on the workshop activities.

A list of open problems.

A collection of papers on the workshop topic.

Papers arising from the workshop:

Truncated Product Representations for $L$-Functions in the Hyperelliptic Ensemble

by J. C. Andrade, S. M. Gonek and J. P. Keating, *Mathematika 64 (2018), no. 1, 137–158 * MR3778218