A characterization of some additive arithmetical functions. V
For Parts I--III see Zbl 0924.11080, Zbl 0969.11033 and Zbl 0969.11034. A real-valued arithmetical function \(f\) is said to be additive if \(f(mn)=f(m)+f(n)\) whenever \((m,n)=1\). \textit{P. Erdős} [Ann. Math. (2) 47, 1-20 (1946; Zbl 0061.07902)] proved that if an additive arithmetical function \(f\) satisfies the condition \(\lim_{n\to\infty}(f(n+1)-f(n))=0\), then \(f(n)=C\log n\) for some constant \(C\). He further conjectured that the conclusion is valid if \(f\) satisfies the condition \(\lim_{x\to\infty}{1\over x}\sum_{n\leq x}|f(n+1)-f(n)|=0\). This was proved by \textit{I. Kátai} [J. Number Theory 2, 1-6 (1970; Zbl 0188.34201)]. These were practically the first of the many articles related to the characterization of the logarithm as an additive arithmetical function. The present article gives a new characterization of the logarithm as an additive arithmetical function.
scientific article; zbMATH DE number 1912069