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Level totients for integer mosaics
Abstract
We study a level analog of Euler's totient function for integer mosaics. Let Pi(n) be the set of primes appearing in the first i levels of the mosaic of n, and let φi(n) count the integers m≤ n for which Pi(m)∩ Pi(n)=∅. We prove a Möbius divisor-sum formula for φi(n) and reduce it to a sum over a set 𝒱i,S of powerful integers. If S≠∅, i≥ 2, and q=min S, then |𝒱i,S∩[1,x]|∼ Ci,Sx1/q, with Ci,S an explicit positive Euler-product constant. For fixed S, the density δi(S) of integers whose first i levels avoid S exists and has an Euler product; for nonempty S, i≥2, and q=min S, the number of such integers up to N is δi(S)N+Oi,S(N1/q). Taking S=Pi(n) gives φi(n)/n=δi(Pi(n))+Oε(n−1/2+ε) uniformly in n.
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BibTeX
@misc{dao2026level,
title = {{Level totients for integer mosaics}},
author = {The Anh Dao and Tim Kasian and Mostafa Mirabi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.22842},
eprint = {2606.22842},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.22842},
}
RIS
TY - UNPB
TI - Level totients for integer mosaics
AU - The Anh Dao
AU - Tim Kasian
AU - Mostafa Mirabi
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2606.22842
UR - https://arxiv.org/abs/2606.22842
AB - We study a level analog of Euler's totient function for integer mosaics. Let P_i(n) be the set of primes appearing in the first i levels of the mosaic of n, and let φ_i(n) count the integers m≤ n for which P_i(m)∩ P_i(n)=∅. We prove a Möbius divisor-sum formula for φ_i(n) and reduce it to a sum over a set 𝒱_i,S of powerful integers. If S≠∅, i≥ 2, and q=min S, then |𝒱_i,S∩[1,x]|∼ C_i,Sx^1/q, with C_i,S an explicit positive Euler-product constant. For fixed S, the density δ_i(S) of integers whose first i levels avoid S exists and has an Euler product; for nonempty S, i≥2, and q=min S, the number of such integers up to N is δ_i(S)N+O_i,S(N^1/q). Taking S=P_i(n) gives φ_i(n)/n=δ_i(P_i(n))+O_ε(n^−1/2+ε) uniformly in n.
ER -