14. Two arithmetic functions $f$ and $g$ may be multiplied using the Dirichlet product which is defined by
$$(f * g)(n)=\sum_{d \mid n} f(d) g(n / d)$$
a) Show that $f^{*} g=g^{*} f$.
b) Show that $\left(f^{*} g\right) * h=f *\left(g^{*} h\right)$.
c) Show that if $\iota$ is the multiplicative function defined by
$$(n)=\left\{\begin{array}{ll} 1 & \text { if } n=1 \\ 0 & \text { if } n>1 \end{array}\right.$$
then $\iota^{*} f=f^{*} \iota=f$ for all arithmetic functions $f$.
d) The arithmetic function $g$ is said to be the inverse of the arithmetic function $f$ if $f^{*} g=g^{*} f=\iota$. Show that the arithmetic function $f$ has an inverse if and only if $f(1) \neq 0$. Show that if $f$ has an inverse it is unique. (Hint: When $f(1) \neq 0$, find the inverse $f^{-1}$ of $f$ by calculating $f(n)$ recursively, using the fact that $\left.\iota(n)=\sum_{d \mid n} f(d) f^{-1}(n / d).\right)$