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Möbius function and prime numbers

  1. Sep 15, 2009 #1
    Let [tex]p_i[/tex] denote the i-th prime number. Prove or disprove that:
    [tex]1)\quad \displaystyle S(n) : = \sum_{i = 1}^n \mu(p_i + p_{i + 1}) < 0 \quad \forall n \in \mathbb{N}_0 : = \left\{1,2,3,...\right\};[/tex]
    [tex] 2)\quad \displaystyle S(n) \sim C \frac {n}{\log{n}},[/tex]
    where [tex]C[/tex] is a negative real constant.

    In the graph attached are represended the functions [tex]-S(n)[/tex] (red)
    and [tex]f (x) : = 0.454353 * x / \log{x}[/tex] (green), with [tex]n,x \in [1, 3 \cdot 10^6].[/tex]

    Attached Files:

  2. jcsd
  3. Sep 15, 2009 #2


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    The pattern holds up to 4e9 - 21, where S(189961812) = -4681611, which suggests C ~= -0.46979.

    But somehow with the failure of the Mertens conjecture I would expect this one to fail eventually.
  4. Sep 16, 2009 #3
    Hi CRGreathouse, thanks for your reply.
    Just one question: how is the trend of [tex]C(n)[/tex] in the variation of [tex]n[/tex]?
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