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1	Mean Square Estimate for Primitive Lattice Points in Convex Planar Domains Coatney, Ryan D. 08 March 2011 (has links) (PDF) 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. Number Theory Lattice Points Hlawka Zeta Function Mean Square Estimate Gauss Circle Problem Riemann Hypothesis Mathematics
2	Lattice Point Counting through Fractal Geometry and Stationary Phase for Surfaces with Vanishing Curvature Campolongo, Elizabeth Grace 02 September 2022 (has links) No description available. Mathematics Heisenberg norm Heisenberg group Heisenberg sphere vanishing curvature lattice point counting stationary phase oscillatory integrals energy integrals Hausdorff dimension Fourier transform surface measure decay fractal geometry geometric measure theory harmonic analysis Fourier analysis intersection of surfaces Gauss circle problem

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Mean Square Estimate for Primitive Lattice Points in Convex Planar Domains

Lattice Point Counting through Fractal Geometry and Stationary Phase for Surfaces with Vanishing Curvature