The number of lattice points within a contour and visible from the origin

Douglas Austin Hensley

Vol. 166 (1994), No. 2, 295–304

DOI: 10.2140/pjm.1994.166.295

Abstract

The main result is an estimate for the number P(r) of relatively prime pairs (a,b) of integers within a contour. When specialized to the contour x² + y² = r this estimate gives

A similar estimate, with the same sort of error, is obtained for the number of relatively prime pairs (a,b) of positive integers so that ab ≤ r. The error term for a general contour depends on the maximal value of the radius of curvature of the bounding contour.

Mathematical Subject Classification 2000

Primary: 11P21

Milestones

Received: 28 September 1992

Revised: 21 December 1993

Accepted: 17 October 1994

Published: 1 December 1994