The probability of choosing primitive sets

Title:
The probability of choosing primitive sets
Authors:
Woods, Kevin; Elizalde, Sergi
Abstract:
We generalize a theorem of Nymann that the density of points in ZdZd that are visible from the origin is 1/ζ(d)1/ζ(d), where ζ(a)ζ(a) is the Riemann zeta function View the MathML source∑i=1∞1/ia. A subset S⊂ZdS⊂Zd is called primitive if it is a ZZ-basis for the lattice Zd∩spanR(S)Zd∩spanR(S), or, equivalently, if S can be completed to a ZZ-basis of ZdZd. We prove that if m points in ZdZd 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))1/(ζ(d)ζ(d−1)⋯ζ(d−m+1)).
Citation:
Woods, Kevin, and Sergi Elizalde. 2007. "The probability of choosing primitive sets." Journal Of Number Theory 125(1): 39-49.
Publisher:
Elsevier for Academic Press
DATE ISSUED:
2007
Department:
Mathematics
Type:
article
PUBLISHED VERSION:
10.1016/j.jnt.2006.11.001
PERMANENT LINK:
http://hdl.handle.net/11282/310375

Full metadata record

DC FieldValue Language
dc.contributor.authorWoods, Kevinen_US
dc.contributor.authorElizalde, Sergien_US
dc.date.accessioned2013-12-23T16:31:44Z-
dc.date.available2013-12-23T16:31:44Z-
dc.date.issued2007en
dc.identifier.citationWoods, Kevin, and Sergi Elizalde. 2007. "The probability of choosing primitive sets." Journal Of Number Theory 125(1): 39-49.en_US
dc.identifier.issn0022-314Xen_US
dc.identifier.urihttp://hdl.handle.net/11282/310375-
dc.description.abstractWe generalize a theorem of Nymann that the density of points in ZdZd that are visible from the origin is 1/ζ(d)1/ζ(d), where ζ(a)ζ(a) is the Riemann zeta function View the MathML source∑i=1∞1/ia. A subset S⊂ZdS⊂Zd is called primitive if it is a ZZ-basis for the lattice Zd∩spanR(S)Zd∩spanR(S), or, equivalently, if S can be completed to a ZZ-basis of ZdZd. We prove that if m points in ZdZd 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))1/(ζ(d)ζ(d−1)⋯ζ(d−m+1)).en_US
dc.publisherElsevier for Academic Pressen_US
dc.identifier.doi10.1016/j.jnt.2006.11.001-
dc.subject.departmentMathematicsen_US
dc.titleThe probability of choosing primitive setsen_US
dc.typearticleen_US
dc.identifier.journalJournal Of Number Theoryen_US
dc.identifier.volume125en_US
dc.identifier.issue1en_US
dc.identifier.startpage39en_US
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