Exercise 3, § 4

Let $f$ be a non-decreasing mapping from $\mathbf{N}$ into $\mathbf{R}$, such that $f(mn) = f(m) + f(n)$ for every pair of relatively prime integers. a) For every integer $a > 1$, and every integer $k > 0$, put

Prove that one has $f(R_k(a)) \geq k f(a), f(S_k(a)) \leq k f(a)$. b) Deduce from a) that for every integer $n > 1$, one has (consider the integer $r \geq 0$ such that $a^r < n \leq a^{r+1}$). c) Conclude that there exists a constant $c \geq 0$ such that $f(n) = c \log n$ for $n \geq 1$. (Observe that $a$ and $n$ are arbitrary in b), and deduce that $f(a)/\log a$ is constant).