On the pointwise periodicity of multiplicative and additive functions

TL;DR

This paper investigates the behavior of multiplicative and additive functions on integers, providing bounds on the number of points where these functions coincide, based on their periodicity and growth rates.

Contribution

It introduces new lower bounds for coincidence points of such functions, depending on their period length and growth rate variations.

Findings

Derived bounds depend on the length of the period.

Growth rate ratios influence the number of coincidence points.

Results apply to idealized gap sets on integers.

Abstract

We study the problem of estimating the number of points of coincidences of an idealized gap on the set of integers under a given multiplicative function $g:\mathbb{N}\longrightarrow \mathbb{C}$ respectively additive function $f:\mathbb{N}\longrightarrow \mathbb{C}$. We obtain various lower bounds depending on the length of the period, by varying the worst growth rates of the ratios of their consecutive values.