Mean Values and Normal Order of a Successive-Iterate Ratio of Dedekind's Arithmetic Function
Year: 2026
Type: preprint
Abstract: We obtain asymptotic formulas for the three sums \[ \sum_{n\le x}\frac{ψ(ψ(n))}{ψ(n)},\qquad \sum_{n\le x}\frac{ψ(n)}{ψ(ψ(n))},\qquad \sum_{n\le x}\log\frac{ψ(ψ(n))}{ψ(n)}, \] where \(ψ\) denotes Dedekind's arithmetic function. We also determine the normal order of the successive-iterate ratio; more precisely, \[ \frac{ψ(ψ(n))}{ψ(n)} \sim \frac{6e^γ}{π^2}\log_3 n \] for almost all positive integers \(n\).
Source: arXiv (Cornell University)
Author: Guo, Aimin
Institution: Anhui Normal University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212011917