
Sara Kropf
Academia Sinica

Stephan Wagner
Stellenbosch University
Keywords:
$q$Additive function, $q$Quasiadditive function, $q$Regular function, Central limit theorem
Abstract
In this paper, we introduce the notion of $q$quasiadditivity of arithmetic functions, as well as the related concept of $q$quasimultiplicativity, which generalise 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.