Nonvanishing at the critical point of the Dedekind zeta functions of cubic $S_3$-fields
ALGEBRAIC GEOMETRY & NUMBER THEORY SEMINAR
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: Can this zeta function vanish at the critical point $s=1/2$? In successive works, 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 for any $n\geq 1$.
When $n=1$, $\zeta_K(s)$ is simply the Riemann zeta function, and Riemann himself established the non vanishing of $\zeta(1/2)$. 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 $K$ satisfy $\zeta_K(1/2)\neq 0$, and Soundararajan establishes that this is in fact true for at least $87.5\%$ of fields $K$ in families of quadratic fields.
In this talk, I will discuss joint work with Anders Södergren 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 critical value.