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I have to prove for $n \in \mathbb{N}>1$ with $n=\prod \limits_{i=1}^r p_i^{e_i}$. $f$ is a multiplicative function with $f(1)=1$:

$\sum_{ d \mid n} \mu(d)f(d)=\prod_{i=1}^r (1-f(p_i))$

How I have to start? Are there different cases or can I prove it in general?

Any help would be fine :)

1 Answers 1

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Please see Theorem 2.18 on page $37$ in Tom Apostol's Introduction to analytic number theory book.

The proof goes as follows:

Define $ g(n) = \sum\limits_{d \mid n} \mu(d) \cdot f(n)$

  • Then $g$ is multiplicative, so to determine $g(n)$ it suffices to compute $g(p^a)$. But note that $g(p^a) = \sum\limits_{d \mid p^{a}} \mu(d) \cdot f(d) = \mu(1)\cdot f(1) + \mu(p)\cdot f(p) = 1-f(p)$
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    @Martin: I can't view the page in google books.2011-11-15