The Probability of Choosing Primitive Sets

Abstract

We generalize a theorem of Nymann that the density of points in Z^d that are visible from the origin is 1/zeta(d), where zeta(a) is the Riemann zeta function 1/1^a + 1/2^a + 1/3^a + ... A subset S of Z^d is called primitive if it is a Z-basis for the lattice composed of the integer points in the R-span of S, or, equivalently, if S can be completed to a Z-basis of Z^d. We prove that if m points in Z^d 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)...zeta(d-m+1)].