  We say that two arithmetic functions f and g form a Mobius 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 Mobius inversion formula of elementary
number theory. In a previous paper, the first-named author showed that if the
members f and g of a Mobius 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 Mobius pair,
either \sum_{n \in supp(f)} 1/n or \sum_{n \in supp(g)} 1/n diverges.
