The Probability of Choosing Primitive Sets

We generalize a theorem of Nymann that the density of points in Z that are visible from the origin is 1/ζ(d), where ζ(a) is the Riemann zeta function P∞ i=1 1/i . A subset S ⊂ Z is called primitive if it is a Z-basis for the lattice Z ∩ spanR(S), or, equivalently, if S can be completed to a Z-basis of Z. We prove that if m points in Z 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) · · · ζ(d−m + 1)).