Numerical Investigation of the Properties of Remainder in Gauss’s Circle Problem

Abstract Results of a numerical experiment on the investigation of the remainder in the problem about the number of integer points in a disk are presented. The pattern of behavior of large deviations of the remainder magnitude from zero is obtained. A numerical confirmation of the hypothesis on the width of maxima according to which all large local maxima of the remainder are fairly wide is obtained, and a hypothetical bound on the remainder magnitude is built. A theorem relating the height (value) of a remainder maximum with the width of this maximum is proved.