Algebra & Number Theory 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 $f g : ℕ \to 𝕋$ , where $𝕋$ denotes the unit circle, such that

\bar{{(f (n), g (n + 1))}_{n \geq 1}} \neq 𝕋 \times 𝕋 .

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

Keywords

multiplicative functions, Erdos discrepancy problem, Katai conjecture

Mathematical Subject Classification 2010

Primary: 11N37

Secondary: 11N64

Milestones

Received: 24 January 2019

Revised: 3 July 2019

Accepted: 5 August 2019

Published: 15 March 2020

Authors

Oleksiy Klurman
Department of Mathematics Royal Institute of Technology Stockholm Sweden

Alexander P. Mangerel
Centre Recherches Mathematiques Université de Montréal Canada