Further generalisations of a classical theorem of Daboussi
Volume 162 / 2020
Colloquium Mathematicum 162 (2020), 109-119
MSC: Primary 11L07; Secondary 11N37.
DOI: 10.4064/cm7961-10-2019
Published online: 10 April 2020
Abstract
According to a well known theorem of Hédi Daboussi, if $\mathcal {M}_1$ stands for the set of those complex-valued multiplicative functions $f$ such that $|f(n)|\le 1$ for all positive integers $n$ and if $\alpha $ is an arbitrary irrational number, then $$ \lim _{x\to \infty } \sup _{f\in \mathcal {M}_1} \left | \frac 1x \sum _{n\le x} f(n) \exp \{2\pi i \alpha n\} \right | =0. $$ Given an infinite set $A$ of positive integers, let $A(x)$ stand for its counting function, and let $\alpha $ be an arbitrary irrational number. We examine various sets $A$ along with an appropriate weight function $w(n)$ for which one can prove that $$ \lim _{x\to \infty } \frac 1{A(x)} \sum _{\substack {n\le x \\ n\in A}} w(n) \exp \{2\pi i \alpha n\} =0. $$
