Some Problems in Number Theory I: the Circle Problem

C.F. Gauss [Gaus1] [Gaus2] showed that the the number of lattice points in circles is the area, πt , up to a tolerance of order at most the circumference, i.e. the square root, O( √ t) . Sierpinski in 1904, following work of Voronoi [Vor] on the Dirichlet Divisor Problem, lowered the exponent in Gauss’s result to 1/3 [Sierp1] [Sierp2]. Further improvements that have been obtained will be described below. This paper provides a proof of the conjectured optimal error bound, O(t 1 4) . The present approach begins with a new asymptotic expression (9) for the number of lattice points in a circle. To obtain such an expression, one begins by embedding the integer lattice within a lattice refined by a factor Q that can be adjusted depending on the size of the circle. A convex polygon with vertices in the refined lattice approximates the circle. This number of lattice points can then be rewritten as a sum of standard trigonometric functions over the points of the refined lattice that lie within this polygon. An elementary generalized Euler-Maclaurin formula (the shortest possible such formula) for lattice sums in lattice polygons then provides an estimate of this sum in terms of certain integrals over the polygon. The lattice polytope structure permits such a formula in terms of Lebesgue measure. The integrals can in turn be approximated by integrals over circular regions; classically, such integrals yield Bessel functions. This procedure thus leads to an expression for the number of lattice points in a circle in terms of sums of Bessel functions or exponential sums. The expressions considered by Hardy, Landau and others have a similar form, but (9) differs in important features; for example, the sum is taken over lattice points in a square, whereas classically the summation pattern in asymptotic formulae for the lattice points in a circle retains the circular symmetry. The classical approach to any such sum next calls for application of Poisson summation, followed by lots of estimates, with the goal of arriving at a situation in which