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TR: [PrimeNumbers] N-th prime boundary?

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  • Didier van der Straten
    To PrimeNumbers groups ... De : Didier van der Straten [mailto:vdstrat@attglobal.net] Envoyé : jeudi 25 décembre 2003 22:17 À : Paul Leyland Objet : RE:
    Message 1 of 3 , Dec 26, 2003
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      To PrimeNumbers groups

      -----Message d'origine-----
      De : Didier van der Straten [mailto:vdstrat@...]
      Envoyé : jeudi 25 décembre 2003 22:17
      À : Paul Leyland
      Objet : RE: [PrimeNumbers] N-th prime boundary?


      Allow me to doubt about this response N/(ln N - 2). This seems to approach
      the "prime count function" up to N, while I believe the question is how to
      approach
      the Nth prime, with a value slightly larger than the actual Nth prime,
      which would suggest a formula such as N * (ln N + k). Finding a "good" k
      might be
      another challenge, including checking it for high values of N.

      Or I obviously missed one point. Forgive me... at Christmas or at New Year's
      eve.
      Didier

      -----Message d'origine-----
      De : Paul Leyland [mailto:pleyland@...]
      Envoyé : mardi 23 décembre 2003 18:05
      À : William Bouris; primenumbers@yahoogroups.com
      Objet : RE: [PrimeNumbers] N-th prime boundary?


      I believe that N/(ln N - 2) will serve your purpose.


      Paul


      > -----Original Message-----
      > From: William Bouris [mailto:melangebillyb@...]
      > Sent: 23 December 2003 16:16
      > To: primenumbers@yahoogroups.com
      > Subject: [PrimeNumbers] N-th prime boundary?
      >
      > Hello, Group.
      >
      > Is there a simple formula for calculating an upper boundary
      > which would always contain a particular nth-prime number?
      > ie. input n --> formula ---> calculated limit > prime p. I
      > bet that this question has been asked several times, but I
      > just need a "slightly larger than p" value for any given n.
      >
      > Thanks in advance, Bill
      >
      >
      >
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    • Edwin Clark
      ... From: bobs@rsa.com Subject: Re: Upper bounds for the nth prime number Date: 1999/03/18 The Rosser & Schoenfeld results, as improved by Robin are the best
      Message 2 of 3 , Dec 26, 2003
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        I found the following sci.math posting which may be of use:

        --------------------------------------------------------------
        From: bobs@...
        Subject: Re: Upper bounds for the nth prime number
        Date: 1999/03/18

        The Rosser & Schoenfeld results, as improved by Robin are the best known.

        p_n > n log n + n (log log n - 1.0072629)

        for n > 7021, p_n <= nlog n + n (log log n - .9385)

        -------------------------------------------------------------

        I don't know a reference for this. Perhaps it can be found in the
        following paper I found on MathSciNet.

        Massias, Jean-Pierre(F-LIMOS-OC); Robin, Guy(F-LIMOS-OC)
        Bornes effectives pour certaines fonctions concernant les nombres
        premiers. (French. French summary) [Effective bounds for certain functions
        involving primes] J. Thor. Nombres Bordeaux 8 (1996), no. 1, 215--242.
        Abstract:
        Let $p_k$ denote the $k$th prime number and let
        $\theta(p_k)=\sum^k_{i=1}\log p_i$ be the Chebyshev function. The authors
        obtain new explicit upper and lower bounds for the expressions $p_k$,
        $\theta(p_k)$, $\sum^k_{i=1}p_i$ and $\sum_{p\leq x}p$, thereby improving
        several estimates obtained by J. B. Rosser and L. Schoenfeld\
        \ref[Illinois J. Math. 6 (1962), 64--94; MR 25 #1139; Math. Comp. 29
        (1975), 243--269; MR 56 #15581a] and Robin \ref[Acta Arith. 42 (1983),
        no. 4, 367--389; MR 85j:11109].
        ----------------------------------------------------------------

        ---Edwin

        ------------------------------------------------------------
        W. Edwin Clark, Math Dept, University of South Florida,
        http://www.math.usf.edu/~eclark/
        ------------------------------------------------------------
      • Jud McCranie
        ... I m not quite sure what you re asking. Are you asking for an upper bound on the nth prime number? A number that is slightly larger than the nth prime?
        Message 3 of 3 , Dec 26, 2003
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          At 05:46 AM 12/26/2003, Didier van der Straten wrote:

          >Allow me to doubt about this response N/(ln N - 2). This seems to approach
          >the "prime count function" up to N, while I believe the question is how to
          >approach
          >the Nth prime, with a value slightly larger than the actual Nth prime,
          >which would suggest a formula such as N * (ln N + k). Finding a "good" k
          >might be
          >another challenge, including checking it for high values of N.

          I'm not quite sure what you're asking. Are you asking for an upper bound
          on the nth prime number? A number that is slightly larger than the nth
          prime? Something else?
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