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Re: small Collection of Primes

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  • djbroadhurst
    ... Since 1923, we have a had a very precise conjecture for the asymptotic density of primes of the form x^2+1. See Shanks review
    Message 1 of 18 , Oct 29, 2012
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      --- In primenumbers@yahoogroups.com,
      "bhelmes_1" <bhelmes@...> wrote:

      > the distribution of primes
      > concerning the polynom f(x)=x^2+1

      Since 1923, we have a had a very precise
      conjecture for the asymptotic density
      of primes of the form x^2+1. See Shanks' review

      http://www.ams.org/journals/mcom/1960-14-072/S0025-5718-1960-0120203-6/S0025-5718-1960-0120203-6.pdf

      of the classic paper by G.H. Hardy and J.E. Littlewood:
      "Some problems of 'Partitio numerorum'; III",
      Acta Math. 44 (1923) pages 1–70.

      The relevant Hardy-Littlewood constant,
      1.3728134... is given, to 9 significant figures,
      in Eq(3) of Shanks' paper.

      More digits are easily obtainable from the methods in
      "High precision computation of Hardy-Littlewood constants"
      by Henri Cohen, available as a .dvi file from
      http://www.math.u-bordeaux1.fr/~hecohen/

      David
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