Some remarks on nonnegative multiplicative functions on arithmetic progressions (Q1281637)

The distribution of general multiplicative functions on arithmetic progressions is a very difficult open problem. In this paper the author considers the simplest case of this problem and establishes an essentially best possible result of the form \[ \sum_{\substack{ n\leq x\\ n\equiv a\pmod q}} \frac{f(n)}{n}\ll \frac{1}{\varphi(q)} \sum_{\substack{ n\leq x\\ (n,q)=1}} \frac{f(n)}{n}, \] where \(f\) is a nonnegative multiplicative function and \((a,q)=1\).