ON COMMON VALUES OF φ(n) AND σ(m), II

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 6 x : n ∈ Vφ ∩ Vσ} 6 Vφ(x) + Vσ(x) (log log x)1/2+o(1) .