On common values of ϕ(n) and σ(m), II

Ford, Kevin; Pollack, Paul

doi:10.2140/ant.2012.6.1669

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

On common values of $\phi(n)$ and $\sigma(m)$, II

Kevin Ford and Paul Pollack

Vol. 6 (2012), No. 8, 1669–1696

DOI: 10.2140/ant.2012.6.1669

Abstract

For each positive-integer valued arithmetic function $f$ , let $V_{f} \subset ℕ$ denote the image of $f$ , and put $V_{f} (x) : = V_{f} \cap [1, x]$ and $V_{f} (x) : = # V_{f} (x)$ . Recently Ford, Luca, and Pomerance showed that $V_{ϕ} \cap V_{σ}$ is infinite, where $ϕ$ denotes Euler’s totient function and $σ$ is the usual sum-of-divisors function. Work of Ford shows that $V_{ϕ} (x) ≍ V_{σ} (x)$ as $x \to \infty$ . Here we prove a result complementary to that of Ford et al. by showing that most $ϕ$ -values are not $σ$ -values, and vice versa. More precisely, we prove that, as $x \to \infty$ ,

# {n \leq x : n \in V_{ϕ} \cap V_{σ}} \leq \frac{V_{ϕ} (x) + V_{σ} (x)}{{(log log x)}^{1 ∕ 2 + o (1)}} .

Keywords

Euler function, totient, sum of divisors

Mathematical Subject Classification 2010

Primary: 11N37

Secondary: 11N64, 11A25, 11N36

Milestones

Received: 29 November 2010

Revised: 30 November 2011

Accepted: 30 January 2012

Published: 14 December 2012

Authors

Kevin Ford
Department of Mathematics University of Illinois 1409 West Green Street Urbana, IL 61801 United States

Paul Pollack
Department of Mathematics University of Georgia Boyd Graduate Studies Research Center Athens, GA 30602 United States

Vol. 6, No. 8, 2012