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More than a simple coincidence?

  1. Apr 8, 2007 #1

    tpm

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    more than a simple coincidence??

    I've been reading Apostol's "Introduction to Analytic Number theory", the chapter dealing with the average of Arithmetical function:

    [tex] \sum_{n \le x} d(n) = xlog(x)+(2 \gamma -1)x+O(\sqrt x ) [/tex]

    [tex] \sum_{n \le x} \sigma _{1} (n) = \frac{ \zeta (2)}{2}x^{2}+O(xlog(x) ) [/tex]

    [tex] \sum_{n \le x} \phi (n) = \frac{ 2}{\zeta (2)}x^{2}+O(xlog(x) ) [/tex]

    Also we know that if RH holds then [tex] \Psi (x) = x+O(\sqrt x) [/tex]

    ain't the coincidence too much remarkable ??..the average values are 'exact' no matter if RH holds or not, using the Dirichlet generatin functions and Perron formula for getting the average orders, you can check that the leading part dominant as x-->oo is given by the contribution of the 'poles' at s=1, x=2 for the generating functions of each arithmetical functions since:

    [tex] \zeta (s) \zeta (s) = s\sum_{n=1}^{\infty}d(n) n^{-s} [/tex]

    [tex] \frac{\zeta (s-1)}{ \zeta (s)} =s \sum_{n=1}^{\infty}\phi (n) n^{-s} [/tex]

    [tex] \zeta (s) \zeta (s-1) = s\sum_{n=1}^{\infty}\sigma_{1} (n) n^{-s} [/tex]

    As it can be seen for every average order always appear at least any of the factors [tex] x^{1/2} [/tex] or log(x) is this a possible hint that RH is true?..we must take into account that for every arithmetical function a sum of the form:

    [tex] \sum _{\rho} a( \rho ) x^{\rho} / \rho [/tex] appears.
     
    Last edited: Apr 8, 2007
  2. jcsd
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