The Gauss Circle Problem

Alan Chang, Princeton University
Fine Hall 314

How many lattice points are contained in the circle of radius R centered at the origin? Gauss used elementary geometric arguments to show that the answer is approximately πR^2. He was able to bound the error of this estimate by O(R). Using some techniques from Fourier analysis and some properties of oscillatory integrals, we can decrease the exponent of R in the error from 1 to 2/3; this will be the main subject of this talk. We will also discuss some ideas that have been used in attempts to lower the exponent further.