## A remark on additive arithmetical functions.(English)Zbl 0159.05802

Generalizing two results of P. Erdős [Ann. Math. (2) 47, 1–20 (1946; Zbl 0061.07902)] the author proves that if $$\varepsilon(n)$$ is an arbitrary sequence tending to zero, and $$f(n)$$ is additive such that $$(f(n+1)-f(n))\geq -\varepsilon(n)$$ then $$f(n)=c\log n$$. The author notes that this result was stated without proof by P. Erdős [Rend. Sem. Mat. Fis. Milano 27, 45–49 (1958; Zbl 0081.04205)].

### MSC:

 11N64 Other results on the distribution of values or the characterization of arithmetic functions