Algebra & Number Theory

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

Kevin Ford and Paul Pollack

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For each positive-integer valued arithmetic function f, let Vf denote the image of f, and put Vf(x):=Vf[1,x] and Vf(x):=#Vf(x). Recently Ford, Luca, and Pomerance showed that VϕVσ 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

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

Received: 29 November 2010
Revised: 30 November 2011
Accepted: 30 January 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 11N64: Other results on the distribution of values or the characterization of arithmetic functions 11A25: Arithmetic functions; related numbers; inversion formulas 11N36: Applications of sieve methods

Euler function totient sum of divisors


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.

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