## Algebra & Number Theory

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

#### Abstract

For each positive-integer valued arithmetic function $f$, let $Vf⊂ℕ$ denote the image of $f$, and put $Vf(x):=Vf∩[1,x]$ and $Vf(x):=#Vf(x)$. Recently Ford, Luca, and Pomerance showed that $Vϕ∩Vσ$ is infinite, where $ϕ$ denotes Euler’s totient function and $σ$ is the usual sum-of-divisors function. Work of Ford shows that $Vϕ(x)≍Vσ(x)$ as $x→∞$. 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 $x→∞$,

$# { n ≤ x : n ∈ V ϕ ∩ V σ } ≤ V ϕ ( x ) + V σ ( x ) ( log log x ) 1 ∕ 2 + o ( 1 ) .$

#### Article information

Source
Algebra Number Theory, Volume 6, Number 8 (2012), 1669-1696.

Dates
Revised: 30 November 2011
Accepted: 30 January 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513729912

Digital Object Identifier
doi:10.2140/ant.2012.6.1669

Mathematical Reviews number (MathSciNet)
MR3033524

Zentralblatt MATH identifier
1279.11093

#### Citation

Ford, Kevin; Pollack, Paul. On common values of $\phi(n)$ and $\sigma(m)$, II. Algebra Number Theory 6 (2012), no. 8, 1669--1696. doi:10.2140/ant.2012.6.1669. https://projecteuclid.org/euclid.ant/1513729912

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