The probability of choosing primitive sets Original Research Article

We generalize a theorem of Nymann that the density of points in image that are visible from the origin is 1/ζ(d), where ζ(a) is the Riemann zeta function image. A subset image is called primitive if it is a image-basis for the lattice image, or, equivalently, if S can be completed to a image-basis of image. We prove that if m points in image are chosen uniformly and independently at random from a large box, then as the size of the box goes to infinity, the probability that the points form a primitive set approaches 1/(ζ(d)ζ(d−1)cdots, three dots, centeredζ(d−m+1)).