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Primorial, (n#)

  1. Oct 17, 2006 #1
    I only am wondering about the Primorial function, n#, (product of all primes less than or equal to n)

    The gamma/factorial function has a nice recursive relationship that is composed of elementary functions; does there exsist an extension to the primorial function?
     
  2. jcsd
  3. Oct 18, 2006 #2

    shmoe

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    There's a simple asymptotic for it's logarithm given by one form of the prime number theorem, log(n#)~n.
     
  4. Nov 10, 2006 #3
    If you define the Chebyshev function:

    [tex] \theta (x)= \sum_{p<x} log(p) [/tex] then:

    [tex] \theta (p_{n}) = log(p#) [/tex] but using this definition the PNT gives

    [tex] log(p # ) \sim nlogn [/tex]
     
  5. Nov 13, 2006 #4

    CRGreathouse

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    p# is about [itex]e^p[/itex]. Pierre Dusart has a paper with fairly tight bounds for this and other functions relating to prime counting.
     
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