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Evaluation of a number-theoretical sum

  1. Aug 9, 2005 #1
    I need some help evaluating the following sum.

    Let [tex] M [/tex] be some large integer

    [tex]Q = \prod_{\substack{p \le \frac{M}{3} \\ (p,M)=1}}{p}[/tex]

    [tex]\mu(n)[/tex] be the Möbius function


    [tex]r(n)=\left\{\begin{array}{cc}1,&\mbox{ if }
    n \ is \ prime\\0, & \mbox{ } otherwise\end{array}\right.[/tex]

    Then I am interested in the sum

    [tex]\sum_{\substack{d|Q \\ d < M}}{\frac{\mu(d)}{d}\sum_{\substack{e|\frac{Q}{d} \\ e(1+2r(e)) < M - d (1 + 2r(d)) }}{\frac{\mu(e)}{e}}}[/tex]

    The related sum
    [tex]\sum_{\substack{e|Q \\ e < x}}{\frac{\mu(e)}{e}}}[/tex]

    is evaluated in www3.sympatico.ca/robert.juricevic/sieve.ps
    as lemma 3.12 on page 25.

    Any help would be greatly appreciated
  2. jcsd
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