1. ## 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

2. (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)