Abstract

The Gauss circle problem in classical number theory concerns the estimation of N(x) = { (m1;m2) in ZxZ : m1^2 + m2^2 <= x }, the number of integer lattice points inside a circle of radius sqrt(x). Gauss showed that P(x) = N(x)- pi * x satisfi es P(x) = O(sqrt(x)). Later Hardy and Landau independently proved that P(x) = Omega_(x1=4(log x)1=4). It is conjectured that inf{e in R : P(x) = O(x^e )}= 1/4. I. K atai showed that the integral from 0 to X of |P(x)|^2 dx = X^(3/2) + O(X(logX)^2). Similar results to those of the circle have been obtained for regions D in R^2 which contain the origin and whose boundary dD satis fies suff cient smoothness conditions. Denote by P_D(x) the similar error term to P(x) only for the domain D. W. G. Nowak showed that, under appropriate conditions on dD, P_D(x) = Omega_(x1=4(log x)1=4) and that the integral from 0 to X of |P_D(x)|^2 dx = O(X^(3/2)). A result similar to Nowak's mean square estimate is given in the case where only "primitive" lattice points, {(m1;m2) in Z^2 : gcd(m1;m2) = 1 }, are counted in a region D, on assumption of the Riemann Hypothesis.

Degree

College and Department

Physical and Mathematical Sciences; Mathematics

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