Uncertainty principles connected with the Möbius inversion formula

We say that two arithmetic functions $f$ and $g$ form a \emph{M\"{o}bius pair} if $f(n) = \sum_{d \mid n} g(d)$ for all natural numbers $n$. In that case, $g$ can be expressed in terms of $f$ by the familiar M\"{o}bius inversion formula of elementary number theory. In a previous paper, the first-named author showed that if the members $f$ and $g$ of a M\"{o}bius pair are both finitely supported, then both functions vanish identically. Here we prove two significantly stronger versions of this uncertainty principle. A corollary is that in a nonzero M\"{o}bius pair, one cannot have both $\sum_{f(n) \neq 0}\frac{1}{n} <\infty$ and $\sum_{g(n) \neq 0}\frac{1}{n} <\infty$.