Mean Values and Normal Order of a Successive-Iterate Ratio of Dedekind's Arithmetic Function

We obtain asymptotic formulas for the three sums $$\sum _{n\le x}\frac{\psi (\psi (n))}{\psi (n)},\sum _{n\le x}\frac{\psi (n)}{\psi (\psi (n))},\sum _{n\le x}\log \frac{\psi (\psi (n))}{\psi (n)},$$ where $\psi$ denotes Dedekind’s arithmetic function. We also determine the normal order of the successive-iterate ratio; more precisely, $$\frac{\psi (\psi (n))}{\psi (n)}\sim \frac{6e^{\gamma }}{{\pi }^2}{\log }_{3}n$$ for almost all positive integers $n$.