UNCERTAINTY PRINCIPLES CONNECTED WITH THE MÖBIUS INVERSION FORMULA

PAUL POLLACK, CARLO SANNA
2013 Bulletin of the Australian Mathematical Society  
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
more » ... ple. 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.
doi:10.1017/s0004972712001128 fatcat:5u2y2psqmrg77h3yvajcykn3ny