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Math Help - Last two proofs

  1. #1
    Junior Member
    Joined
    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 f be a multiplicative function. Prove that

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

    (2) Let f and g be arithmetic functions with g(n)\in \mathbb{R}^+ and f(n)= \prod_{d|n}g(d). for all n \in \mathbb{Z}^+.
    Prove that  g(n)= \prod_{d|n}f(d)^{\mu\left(\frac{n}{d}\right)} for all 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 <br />
F\left( n \right)<br />
is multiplicative, so is <br />
\sum\limits_{\left. d \right|n} {F\left( d \right)} <br />

    In this case let: <br />
F\left( n \right) = \mu \left( n \right) \cdot n<br />
(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: <br />
\ln \left[ {f\left( n \right)} \right] = \sum\limits_{\left. d \right|n} {\ln \left[ {g\left( d \right)} \right]} <br />

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