Uncertainty principles connected with the M\"{o}bius inversion formula

TL;DR

This paper explores uncertainty principles related to the Möbius inversion formula, proving that nontrivial pairs have divergent sums over their supports, extending previous results on finitely supported functions.

Contribution

It establishes stronger uncertainty principles for Möbius pairs, showing that nonzero pairs must have divergent sums over their supports, unlike finitely supported pairs.

Findings

Nonzero Möbius pairs have divergent sums over their supports.

Finitely supported Möbius pairs must be trivial (vanish identically).

Stronger versions of the uncertainty principle are proved.

Abstract

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.