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Mertens Theorem

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  • praxismax
    No response from Brian to my private email, putting to him the last question I posted here :(. If anyone else can help with that Riemann equivalence, I m all
    Message 1 of 1 , May 13, 2012
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      No response from Brian to my private email, putting to him the last question I posted here :(. If anyone else can help with that Riemann equivalence, I'm all ears. Perhaps it follows trivially from the Dirichlet series for phi(n), as mentioned here:
      http://en.wikipedia.org/wiki/Euler%27s_totient_function?
      Isee there's also a potential lead from the 1983 J. L. Nicolas proof mentioned on this page:
      http://en.wikipedia.org/wiki/Riemann_hypothesis#Growth_of_Euler.27s_totient

      Here's another question.

      Can anyone here confirm that the reference

      Mertens, F. "Ein Beitrag zur analytischen Zahlentheorie." J. reine angew. Math. 78, 46-62, 1874

      is the paper that gives the explicit proof of Mertens' Third Theorem? I got the reference from Wolfram:
      http://mathworld.wolfram.com/MertensTheorem.html
      but it is not directly referenced in Wolfram's discussion.

      Incidentally, do you think it's just a matter of igorance that the Wolfram author writes:
      "a full proof of the reversal of the inequality for terms in the Mertens theorem does not seem to appear anywhere in the published literature"? Wikipedia cites a 1983 proof by Guy Robin, and I have identified another, of 2009, by Diamond and Pintz
      http://jtnb.cedram.org/item?id=JTNB_2009__21_3_523_0 ;
      or perhaps the Wolfram author feels that the proofs don't bear scrutiny? Or have I misuderstood something?

      With thanks in advance.
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