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Sandro Bettin & Sary Drappeau: The distribution of the Estermann function and other quantum modular forms

Tid: Må 2021-03-01 kl 17.00 - 18.00

Plats: Zoom, meeting ID: 921 756 1880

Medverkande: Sandro Bettin (University of Genova) & Sary Drappeau (Aix-Marseille University)

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Abstract

For a rational \(a/q\), the Estermann function is defined as the additive twist of the the square of the Riemann zeta-function,

\(D(s,a/q) = \sum_{n>0} d(n) e^{2\pi i n a/q} n^{-s}.\)

It satisfies a functional equation which encodes Voronoi's summation formula.

It is natural to ask how the central values \(D(1/2,a/q)\) are distributed as the rational a/q varies. In contrast with the case of multiplicative twists of L-functions, \(D(s,a/q)\) does not have an Euler product and thus the usual machinery does not apply. However, we are able to employ the fact that \(D(1/2,a/q)\) is a quantum modular form (there is a certain relation between the values at \(a/q\) and \(q/a\)) to show, using dynamical systems methods, that \(D(1/2,a/q)\) is asymptotically distributed as a Gaussian random variable.