PreprintCC BY 4.0arXiv
Mean Values and Normal Order of a Successive-Iterate Ratio of Dedekind's Arithmetic Function
Abstract
We obtain asymptotic formulas for the three sums ∑n≤ xψ(ψ(n))/ψ(n), ∑n≤ xψ(n)/ψ(ψ(n)), ∑n≤ xlogψ(ψ(n))/ψ(n), where ψ denotes Dedekind's arithmetic function. We also determine the normal order of the successive-iterate ratio; more precisely, ψ(ψ(n))/ψ(n) ∼ 6eγ/π2log3 n for almost all positive integers n.
Related
- Extreme values of derivatives of Dedekind zeta functions on the $1$-line
- Normal Behavior and Periodic Points of a Pseudo-Aliquot Map Associated with the Dedekind Psi Function
- Exceptional sets for compositions involving Euler's function,the divisor-sum function and Dedekind's function
- On Special Arithmetic Gamma Values and Thakur's Gauss Sums: From ∞-adic to v-adic
- Upper bounds for arithmetic functions over polynomial values via Chebotarev--Sato--Tate distributions
BibTeX
@misc{guo2026mean,
title = {{Mean Values and Normal Order of a Successive-Iterate Ratio of Dedekind's Arithmetic Function}},
author = {Aimin Guo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06333},
eprint = {2609.06333},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06333},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mean Values and Normal Order of a Successive-Iterate Ratio of Dedekind's Arithmetic Function
AU - Aimin Guo
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06333
UR - https://arxiv.org/abs/2609.06333
AB - We obtain asymptotic formulas for the three sums ∑_n≤ xψ(ψ(n))/ψ(n), ∑_n≤ xψ(n)/ψ(ψ(n)), ∑_n≤ xlogψ(ψ(n))/ψ(n), where ψ denotes Dedekind's arithmetic function. We also determine the normal order of the successive-iterate ratio; more precisely, ψ(ψ(n))/ψ(n) ∼ 6e^γ/π^2log_3 n for almost all positive integers n.
ER -