$q$-Quasiadditive Functions

TL;DR

This paper introduces and characterizes $q$-quasiadditive and $q$-quasimultiplicative functions, generalizing existing concepts, and establishes a central limit theorem encompassing classical and new examples.

Contribution

It defines $q$-quasiadditivity and $q$-quasimultiplicativity, characterizes these functions, especially for $q$-regular functions, and proves a unifying central limit theorem.

Findings

Many natural examples of these functions are characterized.

Characterizations for $q$-regular functions are provided.

A general central limit theorem is established.

Abstract

In this paper, we introduce the notion of $q$-quasiadditivity of arithmetic functions, as well as the related concept of $q$-quasimultiplicativity, which generalises strong $q$-additivity and -multiplicativity, respectively. We show that there are many natural examples for these concepts, which are characterised by functional equations of the form $f(q^{k+r}a + b) = f(a) + f(b)$ or $f(q^{k+r}a + b) = f(a) f(b)$ for all $b < q^k$ and a fixed parameter $r$. In addition to some elementary properties of $q$-quasiadditive and $q$-quasimultiplicative functions, we prove characterisations of $q$-quasiadditivity and $q$-quasimultiplicativity for the special class of $q$-regular functions. The final main result provides a general central limit theorem that includes both classical and new examples as corollaries.