THE NUMBER OF LATTICE POINTS WITHIN A CONTOUR AND VISIBLE FROM THE ORIGIN

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 co ntour x(2) + y(2) = r this estimate gives P(r)= (6/pi)r + (without the RH, O-epsilon(r(1/2)exp(-(log r)((3/5)+epsilon))) or with the RH O(ep silon)r((51+epsilon)/100)). A similar estimate, with the same sort of error, is obtained for the number of relatively prime pairs (a,b) of p ositive integers so that ab less than or equal to r. The error term fo r a general contour depends on the maximal value of the radius of curv ature of the bounding contour.