Erdős Problem #491

Let \(f : \mathbb{N} \to \mathbb{R}\) be an additive function (so that \(f(ab) = f(a) + f(b)\) whenever \((a, b) = 1\)). If \(|f(n+1) - f(n)| < c\) for some constant \(c\) and all \(n\), then must there exist some \(c'\) such that \(f(n) = c' \log n + O(1)\)? A question of Erdős [Er61, p.237; Er82e, p.65], who had proved that \(f(n) = c' \log n\) under the stronger hypotheses \(f(n+1) - f(n) = o(1)\) or \(f(n+1) \ge f(n)\). The answer is yes, proved by Wirsing [Wi70]. See also `erdos_897.variants.log_growth`.