Multiplicative functions with difference tending to zero (Q2759146)

A well known theorem of \textit{P. Erdős} [Ann. Math. (2) 47, 1--20 (1946; Zbl 0061.07902)] characterizes the real-valued, additive functions \(f\) with the property \(f(n+1)- f(n)\to 0\): they are of the form \(c\log n\). The authors extend this in the following way. NEWLINENEWLINENEWLINETheorem 1: If \(f:\mathbb N\to \mathbb R/\mathbb Z\) is additive and \(\lim_{n\to\infty} (f(n+1)-f(n))= 0\), then \(f(n)= c\log n+\mathbb{Z}\) with some real \(c\). NEWLINENEWLINENEWLINETheorem 2: If \(f:\mathbb N\to \mathbb C\) is multiplicative and \(\lim_{n\to\infty} \frac{f(n+1)} {f(n)}= 1\), then \(f(n)= n^s\), \(s\in \mathbb{C}\). NEWLINENEWLINENEWLINETheorem 3: If \(f:\mathbb N\to \mathbb C\) is multiplicative and \(\lim_{n\to\infty} (f(n+1)-f(n))= 0\), then \(f(n)= n^s\) \((s\in \mathbb C\), \(0\leq \operatorname {Re}s< 1)\) or \(\lim_{n\to\infty} f(n)= 0\). NEWLINENEWLINENEWLINEThe proofs are short and very elegant.