
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
$\displaystyle \sum_{dn}\mu(d)f(d) = \prod_{pn}(1f(p))$
(2) Let $\displaystyle f$ and $\displaystyle g$ be arithmetic functions with $\displaystyle g(n)\in \mathbb{R}^+$ and $\displaystyle f(n)= \prod_{dn}g(d).$ for all $\displaystyle n \in \mathbb{Z}^+$.
Prove that $\displaystyle g(n)= \prod_{dn}f(d)^{\mu\left(\frac{n}{d}\right)}$ for all $\displaystyle n \in \mathbb{Z}^+$
Thanks! I'm working on these last two and I am freeee for the summer :D

(1) Remember that if $\displaystyle
F\left( n \right)
$ is multiplicative, so is $\displaystyle
\sum\limits_{\left. d \rightn} {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 \rightn} {\ln \left[ {g\left( d \right)} \right]}
$
And now apply Möbius' inversion formula ( http://www.mathhelpforum.com/mathhe...nformula.html)