Nonvanishing at the critical point of the Dedekind zeta functions of S_3-cubic fields

Abstract: Let K be a number field, and denote the Dedekind zeta function of K by zeta_K(s). A classical question in number theory is: when does this zeta function vanish at the critical point s=1/2? First Armitage, and then Frohlich, gave examples of number fields  which satisfy zeta_K(s)=0. Conversely, it is believed that certain conditions on K can guarantee the nonvanishing of zeta_K(s) at the critical point. For example, it is believed that zeta_K(s) is never 0 when K is an S_n-number field.

When n=2, there has been amazing progress towards understanding the statistics of zeta_K(1/2). Jutila first proved that infinitely many quadratic fields  satisfy zeta_K(1/2)\neq 0, and Soundararajan establishes that this is in fact true for  at least 80% of fields  in families of quadratic fields. In this talk, I will discuss joint work with Anders Sodergren and Nicolas Templier, in which we study the statistics of zeta_K(1/2) in families of S_3-cubic fields. In particular, we will prove that the Dedekind zeta functions of infinitely many such fields have nonvanishing central value.