Bounds for some sums over rational numbers (Q1381637)

A function \(f\) whose domain is the set of positive rational numbers is said to be multiplicative if \(f(m/n)=f(m)f(1/n)\) for coprime positive integers \(m,n\), and that the functions \(f_1(n)=f(n)\), \(f_2(n)=f(1/n)\) are ordinary multiplicative functions with domain the set of natural numbers. The author considers the sum \(\sum f(qm/n+r)\) over rationals \(m/n\) lying in an interval with \(n\leq x\), and \(q,r\) are rationals allowed to vary with \(x\). The theorem gives a uniform bound for such a sum when the multiplicative function \(f\) satisfies certain growth conditions, which are too complicated to be stated here. The method is based on the reviewer's work [\textit{P. Shiu}, J. Reine Angew. Math. 313, 161-170 (1980; Zbl 0412.10030)], where a uniform bound for sums of ordinary multiplicative functions over arithmetic progressions is given, and the result here can be considered to be a generalisation to multiplicative functions over the rationals.