Abstract

Let $f:\mathbb{Z}\to ℂ$ 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 $\chi$.

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