Distribution of values of L-functions at the edge of the critical strip
by
Youness Lamzouri
Université de Montréal

In 2003, Granville and Soundararajan constructed a random model to study the distribution of the values at s=1 of the family of L-functions of quadratic Dirichlet characters, proving a famous conjecture of Montgomery and Vaughan. For higher degree L-functions, Liu, Royer and Wu followed the same philosophy and constructed an adequate random model to study the distribution of the values at s=1 of the family of holomorphic cusp forms in the weight aspect.

We generalize these results, namely by constructing and studying a large class of random models, which include all the previous ones.

Among new applications, we provide a precise estimate for the distribution of the values at s=1 of the family of k-th symmetric power L-functions of holomorphic cusp forms in the level aspect (assuming the automorphy of these L-functions), using results of Cogdell and Michel on high complex moments of this family. Further we study families of L-functions of automorphic forms in the t-aspect, and of L-functions of quadratic twists of an automorphic form at s=1, and used our construction to get precise estimates for the corresponding distribution function assuming a uniform version of the Sato-Tate conjecture.

Date received: June 14, 2008