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SUMMARY:Sandro Bettin & Sary Drappeau (University of Genova & Université 
 d'Aix-Marseille)
DTSTART:20210301T160000Z
DTEND:20210301T170000Z
DTSTAMP:20260423T023936Z
UID:IML_NT/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/IML_NT/9/">T
 he distribution of the Estermann function and other quantum modular forms<
 /a>\nby Sandro Bettin & Sary Drappeau (University of Genova & Université 
 d'Aix-Marseille) as part of IML Number Theory semester (spring 2021)\n\n\n
 Abstract\nFor a rational a/q\, the Estermann function is defined as the ad
 ditive twist of the\nthe square of the Riemann zeta-function\,\n\nD(s\,a/q
 ) = \\sum_{n>0} d(n) e^{2\\pi i n a/q} n^{-s}.\n\nIt satisfies a functiona
 l equation which encodes Voronoi's summation formula. \n\nIt is natural to
  ask how the central values D(1/2\,a/q) are distributed as the\nrational a
 /q varies.  In contrast with the case of multiplicative twists of\nL-funct
 ions\, D(s\,a/q) does not have an Euler product and thus the usual\nmachin
 ery does not apply. However\, we are able to employ the fact that D\n(1/2\
 ,a/q) is a quantum modular form (there is a certain relation between the\n
 values at a/q and q/a) to show\, using dynamical systems methods\, that D\
 n(1/2\,a/q) is asymptotically distributed as a Gaussian random variable.\n
LOCATION:https://researchseminars.org/talk/IML_NT/9/
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