On the Gauss sphere problemMPIM
by
MPIM, Vivatsgasse, 7 - Lecture Hall
Max Planck Institute for Mathematics
Number theory lunch seminar
The Gauss sphere problem concerns counting integer lattice points inside a three-dimensional ball of radius R. A classical result states that the number of such points is asymptotic to the volume of the ball. A central question in analytic number theory is to determine how small the error term in this approximation can be.
Over the years, successive improvements have been obtained by Chen, Vinogradov, Chamizo and Iwaniec, and Heath-Brown, with the latter establishing an error exponent of 21/16.
In this talk, I will discuss recent joint work with Bryce Kerr in which we improve Heath-Brown's exponent to 55/42. I will also explain how recent progress on zero-free regions for Dirichlet L-functions leads to a further improvement.
The corresponding preprint is expected to be available on arXiv in the next few days.