Vol. 6, No. 8, 2012

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

Kevin Ford and Paul Pollack

Vol. 6 (2012), No. 8, 1669–1696

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 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 V ϕ(x) V σ(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 Vϕ Vσ} V ϕ(x) + V σ(x) (loglogx)12+o(1).

Euler function, totient, sum of divisors
Mathematical Subject Classification 2010
Primary: 11N37
Secondary: 11N64, 11A25, 11N36
Received: 29 November 2010
Revised: 30 November 2011
Accepted: 30 January 2012
Published: 14 December 2012
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