Restoring our family values

3:00 pm, Wednesday, 20th March

Abstract

The term “ratios conjectures” refers to asymptotic formulas for the mean values of quotients of L-functions.  Almost exactly a year after my previous number theory seminar talk, we return to the study of special values of families of L-functions.  This talk will report on some recent joint works with P. Gao on the ratios conjectures for (Dirichlet, Hecke) L-functions associated with fixed order characters to prime conductors.  These conductors form sets of zero density in their rings of integers and averaging over sparse sets is generally difficult in analytic number theory.  I shall present some history of and motivation behind this problem before delving into the proofs, which use the method of multiple Dirichlet series.  The technical details will be kept at a minimally necessary level (just enough for me to tell my story).  In celebration of this joint seminar with combinatorics, the work contains at least one proof by induction.