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Thread: Last two proofs

  1. #1
    Junior Member
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    Apr 2008
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    26

    Last two proofs

    After these next two, I'm done for the semester! Thanks to all!

    (1) Let $\displaystyle f$ be a multiplicative function. Prove that

    $\displaystyle \sum_{d|n}\mu(d)f(d) = \prod_{p|n}(1-f(p))$

    (2) Let $\displaystyle f$ and $\displaystyle g$ be arithmetic functions with $\displaystyle g(n)\in \mathbb{R}^+$ and $\displaystyle f(n)= \prod_{d|n}g(d).$ for all $\displaystyle n \in \mathbb{Z}^+$.
    Prove that $\displaystyle g(n)= \prod_{d|n}f(d)^{\mu\left(\frac{n}{d}\right)}$ for all $\displaystyle n \in \mathbb{Z}^+$

    Thanks! I'm working on these last two and I am freeee for the summer
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  2. #2
    Super Member PaulRS's Avatar
    Joined
    Oct 2007
    Posts
    571
    (1) Remember that if $\displaystyle
    F\left( n \right)
    $ is multiplicative, so is $\displaystyle
    \sum\limits_{\left. d \right|n} {F\left( d \right)}
    $

    In this case let: $\displaystyle
    F\left( n \right) = \mu \left( n \right) \cdot n
    $ (this function is multiplicative since it's the product of two multiplicative functions)

    Try to work with that

    (by multiplicative I mean weakly multiplicative, not completely multiplicative)


    (2) note that: $\displaystyle
    \ln \left[ {f\left( n \right)} \right] = \sum\limits_{\left. d \right|n} {\ln \left[ {g\left( d \right)} \right]}
    $

    And now apply Möbius' inversion formula ( http://www.mathhelpforum.com/math-he...n-formula.html)
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