The probability of choosing primitive sets |
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Authors: | Sergi Elizalde Kevin Woods |
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Institution: | a Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA b Department of Mathematics, Oberlin College, Oberlin, OH 44074, USA |
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Abstract: | We generalize a theorem of Nymann that the density of points in Zd that are visible from the origin is 1/ζ(d), where ζ(a) is the Riemann zeta function . A subset S⊂Zd is called primitive if it is a Z-basis for the lattice Zd∩spanR(S), or, equivalently, if S can be completed to a Z-basis of Zd. We prove that if m points in Zd 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)). |
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Keywords: | Primitive sets Visible points Random lattice points |
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