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