(Electronic Journal of Combinatorics, 2017) Kropf, Sara; Wagner, Stephan
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(qk+ra+b) = f(a)+f(b) or f(qk+ra+b) = f(a)f(b) for all b < qk 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.