## Convolution Question

$\displaystyle \lambda(n) = (-1)^{a1+a2+....+ak}$ where $\displaystyle n = \prod pi^{ai}$ for distinct primes pi
g(n) = 1 if n = 1 or n = $\displaystyle \prod pi$ for distinct primes pi
and g(n) = 0 otherwise

Assume $\displaystyle \lambda$ and g are multiplicative.

a) Prove that g is the inverse of $\displaystyle \lambda$ under the Dirichlet convolution operation.

b) Let $\displaystyle \Lambda = (\lambda * po)$ be the summatory function of $\displaystyle \lambda$. Prove that
$\displaystyle \Lambda(n) = 1$ if n is a perfect square and 0 otherwise

c) Define a collection of arithmetic functions $\displaystyle g^{(m)} : Z^+ \rightarrow \complex$ as follows:
$\displaystyle g^{(1)} = g$
$\displaystyle g^{(m)} = (g * g^{(m-1)})$ for $\displaystyle m \geq 2$
Prove that $\displaystyle g^{(m)}$ is multiplicative for all $\displaystyle m \in Z^+$ and show that for any prime p and $\displaystyle k \in Z^+$,
$\displaystyle g^{(m)}(p^k) = \left(\begin{array}{cc}m\\k\end{array}\right)$ if $\displaystyle 0 \leq k$ $\displaystyle \leq m$
and 0 if k > m

I'm given a hint for (c)... Use mathematical induction