A generalization of a theorem of Ayoub and Chowla

Abstract

Let X 1 \mathcal {X}1 and X 2 \mathcal {X}2 be characters modulo q 1 {q_1} and q 2 {q_2} , respectively, where q 1 {q_1} and q 2 {q_2} are positive integers. Let \[ f ( n ) = ∑ d | n X 1 ( d ) X 2 ( n / d ) . f(n) = \sum \limits _{d|n} \mathcal {X}1 (d)\mathcal {X}2(n/d). \] In this paper we shall give an estimate for the sum \[ ∑ n ⩽ x f ( n ) log ⁡ ( x / n ) . \sum \limits _{n \leqslant x} {f(n)} \log (x/n). \]