# The probability of choosing primitive sets

- Title:
- The probability of choosing primitive sets
- Authors:
- 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:
- 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 Field | Value | Language |
---|---|---|

dc.contributor.author | Woods, Kevin | en_US |

dc.contributor.author | Elizalde, Sergi | en_US |

dc.date.accessioned | 2013-12-23T16:31:44Z | - |

dc.date.available | 2013-12-23T16:31:44Z | - |

dc.date.issued | 2007 | en |

dc.identifier.citation | Woods, Kevin, and Sergi Elizalde. 2007. "The probability of choosing primitive sets." Journal Of Number Theory 125(1): 39-49. | en_US |

dc.identifier.issn | 0022-314X | en_US |

dc.identifier.uri | http://hdl.handle.net/11282/310375 | - |

dc.description.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)). | en_US |

dc.publisher | Elsevier for Academic Press | en_US |

dc.identifier.doi | 10.1016/j.jnt.2006.11.001 | - |

dc.subject.department | Mathematics | en_US |

dc.title | The probability of choosing primitive sets | en_US |

dc.type | article | en_US |

dc.identifier.journal | Journal Of Number Theory | en_US |

dc.identifier.volume | 125 | en_US |

dc.identifier.issue | 1 | en_US |

dc.identifier.startpage | 39 | en_US |

All Items in The Five Colleges of Ohio Digital Repository are protected by copyright, with all rights reserved, unless otherwise indicated.