Uncertainty principles connected with the Möbius inversion formula

Two arithmetic functions f and g form a Möbius pair if f (n) = Σ d g(d) for all natural numbers n. In that case, g can be expressed in terms of f by the familiar Möbius inversion formula of elementary number theory. In a previous paper, the first-named author showed that if the members f and g of a Möbius 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 Möbius pair, one cannot have both Σf (n)#01=n < 1 and Σg(n),01=n < ∞. © 2012 Australian Mathematical Publishing Association Inc.