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Sum over primes

  1. May 21, 2007 #1


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    If we use the Laplace transform analysis applied to:

    [tex] \sum_{p} exp(-sp) = \int_{0}^{\infty}dt \frac{d\pi (t)}{dt}e^{-st} [/tex]

    then using the properties of Laplace transform (i have consulted to "MATHEMATICAL HANDBOOK OF FORMULA AND TABLES" by Spiegel & Avellanas) we find the relations:

    [tex] s^{-n-1} \sum_{p} exp(-p/s) \rightarrow t^{n/2} \sum_{p}p^{-n/2}J_{n} (2\sqrt{tp} [/tex]

    [tex] s^{-1} \sum_{p} exp(-s^{1/2} p) \rightarrow (t \pi )^{-1/2} \sum_{p} exp(-p^{2}/4t) [/tex]

    amazingly the Laplace transform of a sum over primes is just another sum over primes
  2. jcsd
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