On common values of<i>ϕ</i>(<i>n</i>) and<i>σ</i>(<i>m</i>), II

Abstract

On common values of φ(n) and σ (n), II Kevin Ford and Paul PollackFor each positive-integer valued arithmetic function f , let V f ⊂ ‫ގ‬ denote the image of f , and putRecently Ford, Luca, and Pomerance showed that V φ ∩ 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 → ∞.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 → ∞,V φ (x) + V σ (x) (log log x) 1/2+o(1) .

Locations

• Algebra & Number Theory - View - PDF
• Project Euclid (Cornell University) - View - PDF