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Thread: Dirichlet characters

  1. December 8th 2010, 06:38 AM #1
    Capillarian
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    Dirichlet characters

    I've been trying to justify the following step in a textbook (Iwaniec-Kowalski) proof, but just can't seem to:

    \displaystyle \sum_{n\:\equiv\: 0 \mod k} \chi (n) = \sum_{l|k} \mu (l) \sum_{(n,l)=1} \chi (n)

    where \mu is the Mobius function. It's clearly some application of Mobius inversion. Can anyone show this rigorously?
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  2. December 9th 2010, 09:32 AM #2
    chiph588@
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    Hint: \displaystyle \sum_{d\mid n} \mu(d) = \begin{cases} 1 \text{, if } n = 1 \\ 0 \text{, otherwise} \end{cases}
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