Igor Wigman: Introduction to Voronoi type summation, part one - applications to Gauss's circle problem

Roughly speaking, the Voronoi summation approach is a versatile method for obtaining asymtotics of sums provided a certain associated generating function satisfies a functional equation (e.g. Riemann's zeta function.) The purpose of this series of talks is to learn about this powerful method.

In the first talk, we will discuss the first application of the Voronoi type summation formulae for the Gauss circle problem. It is well known that the number of lattice points inside a disc of a large radius t, is asymptotic to the area of the disc. An elementary geometric argument by Gauss in 1837, shows that the remainder term is O(t). Some more sophisticated arguments (Sierpinski, 1904, Van der Corput, 1920 and Vinogradov, 1930s), show that the correct exponent of t in the upper bound may be proved to be at most 2/3. In this talk I plan to discuss the background, as well as present two different arguments for the validity of this statement. First, I will present the Voronoi summation formula approach, which gives a precise formula for the remainder term. This is just one application of an extremely robust and rich subject of research. Next, I will show a modernization of Van der Corput's original approach: working with the smoothed version of the counting function. This results in a more elementary though essentially the same computation.