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DTSTAMP:20260404T154749Z
UID:1622116800@ist.ac.at
DTSTART:20210527T140000
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DESCRIPTION:Speaker: Arul Shankar\nhosted by Tim Browning\nAbstract: Let $K
 $ be a number field\, and denote the Dedekind zeta function of $K$ by $\\z
 eta_K(s)$. A classical question in number theory is: Can this zeta functio
 n vanish at the critical point $s=1/2$?  In successive works\, Armitage\,
  and then Frohlich\, gave examples of number fields  which satisfy $\\zet
 a_K(s)=0$. Conversely\, it is believed that certain conditions on $K$ can 
 guarantee the nonvanishing of $\\zeta_K(s)$ at the critical point. For exa
 mple\, 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 und
 erstanding the statistics of $\\zeta_K(1/2)$. Jutila first proved that inf
 initely many quadratic fields $K$ satisfy $\\zeta_K(1/2)\neq 0$\, and Soun
 dararajan 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 j
 oint 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 pa
 rticular\, we will prove that the Dedekind zeta functions of infinitely ma
 ny such fields have nonvanishing critical value.
LOCATION:https://mathseminars.org/seminar/AGNTISTA\, ISTA
ORGANIZER:birgit.oosthuizen-noczil@ist.ac.at
SUMMARY:Arul Shankar: Nonvanishing at the critical point of the Dedekind ze
 ta functions of cubic $S_3$-fields
URL:https://talks-calendar.ista.ac.at/events/3151
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