Families and statistics of L-functions

Abstract

The first chapter of this dissertation provides a general introduction to the study of families of L-functions along with the necessary tools for understanding their behavior. In particular, we introduce the families studied in the second and third chapters of this dissertation and provide some prerequisite knowledge on these families. The second chapter of this dissertation studies a family of L-functions attached to Hecke Grossencharacters and extends a geometric result of Ricci concerning the equidistribution of prime ideals of Z[i] in narrow sectors. The third chapter of this dissertation studies a family of L-functions attached to automorphic forms on GL₂. Specifically, we investigate the sixth moment of the family of L-functions associated to holomorphic modular forms on GL₂ with respect to a congruence subgroup [gamma]₁(q). We improve on previous work and obtain an unconditional upper bound of the correct order of magnitude.