The distribution of the Estermann function and other quantum modular forms

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.

number theory

Audience: researchers in the topic

IML Number Theory semester (spring 2021)