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Re: Link - Riemann hypothesis link?

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  • andrew_j_walker <ajw01@uow.edu.au>
    ... maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1043940786913_2887&s tored_search=&FIRSTINDEX=20&tocsectionid=Mathematics ... Thanks for posting this, it
    Message 1 of 3 , Feb 1, 2003
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      --- In primenumbers@yahoogroups.com, Jose Ramón Brox <ambroxius@t...>
      wrote:
      > A pseudo zeta function and the distribution of primes:
      >
      > http://www.pnas.org/cgi/content/full/97/14/7697?
      maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1043940786913_2887&s
      tored_search=&FIRSTINDEX=20&tocsectionid=Mathematics
      >
      > Regards. Jose Brox.
      >
      >

      Thanks for posting this, it is indeed very interesting!!
      The statement that caught my eye is a few lines above equation [4].
      In Equation [2] the prime zeta function phi(s) is defined.
      Then before [4] he states:
      " hence the Riemann hypothesis is equivalent to the claim that phi(s)
      can be analytically continued into the strip 1/2 < Re s <1."

      Now as I mentioned in an earlier posting, Carl-Erik Froberg in his
      1968 paper "On The Prime Zeta Function" makes the statement in his
      introduction, after defining P(s) as the prime zeta function:
      "In the next section we shall show that P(s) can be expressed as an
      infinite series involving the usual Riemann zeta function.
      Simultaneously, this formula provides an analytic continuation to the
      strip 0 < sigma <= 1."

      If indeed both of these statements are correct and agree in their
      definition of analytic continuation, this I belive would imply a
      proof of Riemann's hypothesis. Unfortuneately Froberg gives no
      justification of the second sentence, while the first refers to the
      usual mobius and log sum which is given.

      If what I've written above makes sense and isn't all fluff, then I
      would strongly suggest that someone check one of


      Landau-Walfisz, Rend. di Palermo 44 (1919), p. 82-86

      Glaisher, Quart. Jour. of Math. 25 (1891), p. 347

      C.W. Merrifield, The sums of the series of reciprocals of the
      prime numbers and of their powers, Proc. Roy. Soc. London, (1881),
      vol. 33, p. 4-10

      and see if these are where Froberg makes his assertions from. Perhaps
      I should email Froberg directly? (A google search shows he is still
      working in Sweden at at Lund University and Lund Institute of
      Technology)

      Thanks,

      Andrew Walker
    • David Broadhurst <d.broadhurst@open.ac.u
      Andrew: Please do contact Froeberg. And why not also Chernoff? Mind you, I d be very surprised if there is a link from the (partial) analyticity of Froeberg s
      Message 2 of 3 , Feb 1, 2003
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        Andrew: Please do contact Froeberg.
        And why not also Chernoff?
        Mind you, I'd be very surprised if
        there is a link from the (partial)
        analyticity of Froeberg's zeta
        to RH. But it's always good to ask.
        David
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