Alex Popa: On the second moment of Hecke L-functions of r-th order characters

We establish asymptotic formulas for the second moment of central

values of $L$-functions

associated with $r$-th order Hecke characters. Our analysis covers two

distinct families:

characters attached to square-free ideals and those attached to $r$-th

power-free ideals.

In the latter case, the asymptotic formula includes secondary-order terms.

As an application we obtain explicit, positive proportions for the

nonvanishing of

central values of these $L$-functions.

 

Our approach relies on the method of multiple Dirichlet series to achieve

the necessary analytic continuation for the generating series for

the second moment. This is joint work with Adrian Diaconu, Bogdan Ion

and Vicentiu Pasol.