Numerical Studies of the Gauss Lattice Problem

The difference between the number of lattice points N(R) that lie in x2 + y2 :s R2 and the area of that circle, d(R) = N(R) 7rR2 , can be bounded by Id(R)1 :S KRIJ . Gauss showed that this holds for () = 1, but the least value for which it holds is an open problem in number theory. We have sought numerical evidence by tabulating N(R) up to R:::::: 55,000. From the convex hull bounding log Id(R)1 versus logR we obtain the bound () :S 0.575 , which is significantly better than the best analytical result () :S 0.6301 ... due to Huxley. The behavior of d(R) is of interest to those studying quantum chaos. * This research was supported in part by the National Science Foundation under Cooperative Agreement No. CCR-912000B. The government has certain rights in this material. Numerical Studies of the Gauss Lattice Problem H. B. Keller California Institute of Technology Pasadena, CA 91125, USA