# Last two proofs

• May 1st 2008, 10:28 AM
Proof_of_life
Last two proofs
After these next two, I'm done for the semester! Thanks to all!

(1) Let $f$ be a multiplicative function. Prove that

$\sum_{d|n}\mu(d)f(d) = \prod_{p|n}(1-f(p))$

(2) Let $f$ and $g$ be arithmetic functions with $g(n)\in \mathbb{R}^+$ and $f(n)= \prod_{d|n}g(d).$ for all $n \in \mathbb{Z}^+$.
Prove that $g(n)= \prod_{d|n}f(d)^{\mu\left(\frac{n}{d}\right)}$ for all $n \in \mathbb{Z}^+$

Thanks! I'm working on these last two and I am freeee for the summer :D
• May 1st 2008, 11:14 AM
PaulRS
(1) Remember that if $
F\left( n \right)
$
is multiplicative, so is $
\sum\limits_{\left. d \right|n} {F\left( d \right)}
$

In this case let: $
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: $
\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)