Let H(n) = σ(ϕ(n))/ϕ(σ(n)), where ϕ(n) is Euler's function and σ(n) stands for the sum of the positive divisors of n. We obtain the maximal and minimal orders of H(n) as well as its average order, and we also prove two density theorems. In particular, we answer a question raised by Golomb.
We study the concentration of the distribution of an additive function f when the sequence of prime values of f decays fast and has good spacing properties. In particular, we prove a conjecture by Erdős and Kátai on the concentration of when c > 1.
Given an integer base and a completely -additive arithmetic function taking integer values, we deduce an asymptotic expression for the counting functionunder a mild restriction on the values of . When , the base sum of digits function, the integers counted by are the so-called base Niven numbers, and our result provides a generalization of the asymptotic known in that case.