Note on the derived logarithmic fuction of a multiplicative arithmetic function

Abstract

The $abc-$conjecture on the derived logarithmic function $L_{\varphi}$ of the Euler function was submitted. However, it remains unresolved at present, partly because the details of various properties about $L_{\varphi}$ and the $L_{\varphi}$ value of the radical number are not clear. In this paper, we confirm the experimental results when the Euler function $\varphi$ is replaced by the modified Dedekind $\psi$ function $\psi_{0}$, and show that the derived logarithmic function $L_{\varphi}$ of the Euler function is most likely optimal with respect to the establishment of the $abc-$conjecture for the derived logarithmic function of the multiplication function. Note also the behavior of the evaluation inequality $\log_{3} x \leqq L_{\varphi}(x) \leqq \log_{2}x$ and the modified Dedekind's $\psi$ function $\psi_{0}$.