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Math Help - Liouville's function

  1. #1
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    Liouville's function

    Prove that the Liouvilles function is given by the formula \lambda(n)= \sum\limits_{d^{2} \mid n} \mu \Bigl( \frac{n}{d^2} \Bigr)
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  2. #2
    Super Member PaulRS's Avatar
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    First prove that F(n) = \sum_{d|n}\lambda(d) is 1 if n is a square and 0 otherwise. (you may prove that factoring, notice that Liouville's function is multiplicative)

    Then, by Möbius' Inversion Formula: \lambda(n)=\sum_{d|n}F(d)\cdot \mu\left(\frac{n}{d}\right) but remember what I said above, then \lambda(n)=\sum_{d^2|n}F(d^2)\cdot \mu\left(\frac{n}{d^2}\right)=\sum_{d^2|n}\mu\left  (\frac{n}{d^2}\right) since all the other terms are 0.
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