Density properties of fractions with Euler’s totient function

Halupczok, Karin; Ohst, Marvin

doi:10.2140/involve.2026.19.1

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Density properties of fractions with Euler's totient function

Karin Halupczok and Marvin Ohst

Vol. 19 (2026), No. 1, 1–31

DOI: 10.2140/involve.2026.19.1

Abstract

We prove that, for all constants $a \in ℕ$ , $b \in ℤ$ , $c, d \in ℝ$ , $c \neq 0$ , the fractions $φ (a n + b) ∕ (c n + d)$ lie dense in the interval $] 0, D]$ (and in $[D, 0 [$ if $c < 0$ ), where $D = a φ (\gcd (a, b)) ∕ (c \gcd (a, b))$ . This interval is the largest possible, since it may happen that isolated fractions lie outside of the interval: we prove a complete determination of the case where this happens, which yields an algorithm that calculates the number of $n$ such that $rad (a n + b) | g$ for coprime $a, b$ and any $g$ . Furthermore, this leads to an interesting open question which is a generalization of a famous problem raised by V. Arnold. We prove that the fractions $φ (a n + b) ∕ φ (c n + d)$ with constants $a, c \in ℕ$ , $b, d \in ℤ$ lie dense in $] 0, \infty [$ exactly when $a d \neq b c$ .

Keywords

Euler's totient function, Schinzel density, Arnold's question

Mathematical Subject Classification

Primary: 11A07, 11N37, 11N56, 11N69

Milestones

Received: 27 September 2022

Revised: 9 July 2024

Accepted: 2 September 2024

Published: 17 January 2026

Communicated by Vadim Ponomarenko

Authors

Karin Halupczok
Mathematisches Institut Heinrich Heine Universität Düsseldorf Düsseldorf Germany

Marvin Ohst
Mathematisches Institut Heinrich Heine Universität Düsseldorf Düsseldorf Germany

Vol. 19, No. 1, 2026