The Halász inequality for an additive function of rational argument (Q1881782)

Let \(f\) be an additive function defined on the set of positive rational numbers. The author proves that \[ \left| \left\{f(m/n)=a, n\leq x, \xi <m/n<\eta \right\} \right| \ll x^2 (\xi -\eta ) \biggl( \sum _{p\in P} 1/p \biggr)^{-1/2}, \] where \(P\) denotes the set of primes \(p\) such that \(f(p^\varepsilon )\neq 0\) and \(p^\varepsilon \) occurs in the canonical decomposition of some rational numbers on the left side for some \(\varepsilon =\pm 1\). This is analogous to a theorem of \textit{G. Halász} for integer-valued functions [Acta Arith. 27, 143--152 (1975; Zbl 0256.10028)]. The proof is based on applying Halász' result to the additive functions \(g,h\) defined by \(f(m/n)=g(m)+h(n)\).