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

RESUMO

We obtain asymptotic formulas for the three sums \[ \sum_{n\le x}\frac{\psi(\psi(n))}{\psi(n)},\qquad \sum_{n\le x}\frac{\psi(n)}{\psi(\psi(n))},\qquad \sum_{n\le x}\log\frac{\psi(\psi(n))}{\psi(n)}, \] 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 \(n\).