On a conjecture of Kátai for additive functions
Let \(|x|\) denote the distance of \(x\) from the nearest integer and \(\Delta\) the forward difference of an arithmetic function \(f: \Delta f(n)= f(n+ 1)- f(n)\). The authors give two proofs of Kátai's conjecture: if \(f: \mathbb{N}\to \mathbb{R}\) is additive and \((*)\) \(|\Delta f(n)|\to 0\), then there exists \(\tau\in \mathbb{R}\) such that \[ |f(n)- \tau\log n|= 0\quad \text{for all } n. \] Weaker versions (with stronger conditions than \((*)\)) are already known.
scientific article; zbMATH DE number 874189