Playing with additivity conditions in multiplicative functions

Abstract

Let $f:\mathbb{Z}\to \mathbb{C}$ be a multiplicative function. Assume there exist integers $a>1$ and $d>1$ such that $(a,d)=1$ and let ${\mathcal{P}}_{a,d}=\{a+kd:k\in \mathbb{Z}\}$. Under mild extra conditions on $d$ and $f(a)$, we prove that $f(n)=n\chi (n)$ for all $n$ outside an explicit exceptional set depending on $d$, and some Dirichlet character $\chi$.