The Kubilius inequality for additive functions of a rational argument
A function \(f\) of a rational argument is called additive if it satisfies \(f((a/b)(c/d))=f(a/b)f(c/d)\) whenever \(a, b, c, d\) are pairwise coprime. If \(I=[s,t]\) and \(x>0\), let \(Q_ x\) be the set of numbers \(m/n\in I\) with \((m,n)=1,\;n\leq x\). Let \(\Pi\) be the set of those prime-powers \(p^{\alpha}\) for which there is a \(q\in Q_ x\) whose decomposition contains \(p^{\alpha}\) (here \(\alpha\) may be negative). Write \[ A=\sum_{p^{\alpha}\in \Pi}f(p^{\alpha})p^{-| \alpha |},\quad B=\sum_{p^{\alpha}\in \Pi}f(p^{\alpha})^ 2p^{- | \alpha |}. \] The main result of the paper asserts that \(\sum_{q\in Q_ x}| f(q)-A|^ 2\ll x^ 2(t-s)B,\) where the interval \(I\) may even depend on \(x\), the only constraints are that \(t-s\geq x^{-\mu}\) with some \(0\leq \mu <1\) and \(s\ll x(t-s)\). The implied constant depends on this \(\mu\). The proof proceeds in an unusual way: first an analog of Elliott's dual inequality is proved using the large sieve, and then this is dualized to yield this analog of the Turán-Kubilius inequality. It is unclear whether the familiar direct proofs could be extended to rational functions.
scientific article; zbMATH DE number 4164980