#### Vol. 14, No. 1, 2020

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On the orbits of multiplicative pairs

### Oleksiy Klurman and Alexander P. Mangerel

Vol. 14 (2020), No. 1, 155–189
DOI: 10.2140/ant.2020.14.155
##### Abstract

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

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

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

##### Keywords
multiplicative functions, Erdos discrepancy problem, Katai conjecture
Primary: 11N37
Secondary: 11N64