Define the Euler totient function $\phi$ and the Möbius function $\mu$. Suppose $f$ and $g$ are functions defined on the natural numbers satisfying $f(n)=\sum_{d \mid n} g(d)$. State and prove a formula for $g$ in terms of $f$. Find a relationship between $\mu$ and $\phi$.

Define the Riemann zeta function $\zeta(s)$. Find a Dirichlet series for $\zeta(s-1) / \zeta(s)$ valid for $\operatorname{Re}(s)>2$.