On Products of Multiplicative Functions of Absolute Value at Most 1 Which are Composed with Linear Functions

Abstract

Let

$$\phi (n) = \prod\limits_{j = 1}^q {f_j (a_j n + b_j )}$$

, where a ₁,…, a _q , b ₁,…, b _q are integers with a _j ≥ 1 and a _j + b _j ≥ 1 (so that a _j n + b _j is a positive integer for every n ≥ 1), and f ₁,…, f _q are multiplicative functions of absolute value ≥ 1. We are interested here in the behavior of

$$\frac{1}{x}\sum\limits_{n \leqslant x} \phi (n)$$

when x tends to infinity.