On the orbits of multiplicative pairs

Abstract

We characterize all pairs of completely multiplicative functions $fg:\mathbb{N}\to \mathbb{𝕋}$, where $\mathbb{𝕋}$ denotes the unit circle, such that

$$\overline{{\left\{\left(f\left(n\right),g\left(n+1\right)\right)\right\}}_{n\ge 1}}\ne \mathbb{𝕋}\times \mathbb{𝕋}.$$

In so doing, we settle an old conjecture of Zoltán Daróczy and Imre Kátai.