On total additive solution of some equations (Q1207383)

An arithmetical function \(f\) is said to be additive if (1) \(f(mn)=f(m)+f(n)\) whenever \((m,n)=1\). An additive function is said to be totally additive if (1) holds for all \(m\), \(n\). The logarithm function \(\log n\) is a totally additive function. Since \textit{P. Erdős}' paper [Ann. Math., II. Ser. 47, 1-20 (1946; Zbl 0061.079)] much attention has been paid to the problem: How regular can an additive function be if it is not a logarithm. Erdős proved that if \(f\) is additive and \(f(n+1)- f(n)\to 0\), then \(f(n)=c\log n\). In these results the arguments are usually linear. The author gives some conditions for totally additive functions which contain quadratic arguments or arguments \(n^ k+s\) and which imply \(f(n)=c\log n\).