On some problems of Mąkowski–Schinzel and Erdős concerning the arithmetical functions φ and σ

Abstract

Let σ(n) denote the sum of positive divisors of the integer n, and let φ denote Euler's function, that is, φ(n) is the number of integers in the interval [1, n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(φ(n))/n ≥ 1/2 for all n. We show that σ(φ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(φ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that φ(n − φ(n)) < φ(n) on a set of asymptotic density 1.

Divisors of the Euler and Carmichael functions

“…The distribution of φ(n) and λ(n) has been investigated from a variety of perspectives. In particular, many interesting properties of these functions require knowledge of the distribution of prime factors of φ(n) and λ(n) (see e.g., [3]- [7], [12], [19]). …”

Irrationality of Power Series for Various Number Theoretic Functions

“…Proof. It follows from a much more general statement about divisibility of the values of the Euler function, obtained in the proof of Theorem 4.1 of [2], that the number of positive integers n ≤ T with 3 ϕ(n) is O(T log −1/2 T ) (see also [8]). The same method also extends to integers with 3 σ(n) without any modifications.…”

Values of the Euler Function in Various Sequences

“…By Lemma 2 of [24], we know that as t tends to infinity, the set G(t) of positive integers m ≤ t such that ϕ(m) is divisible by all primes p < log 2 t/(log 3 t) 2 is of cardinality #G(t) = (1 + o(1))t. By the Mertens formula, we see that the inequality…”

“…We also need the following estimate, which is a variant of Lemma 2 of [24] and can be obtained by the same arguments:…”

“…We recall that, for any positive integer n, ϕ(n) is the cardinality of the multiplicative group U n = (Z/nZ) × , while λ(n) is the maximal order of any element in U n . There exists an extensive literature in which the distributional and arithmetical properties of ϕ(n) and λ(n) have been studied (for example, see [1,3,4,5,6,8,9,12,13,14,15,16,17,24,26,28,29]). Here, we list a few examples of properties and interrelations between ϕ(n), λ(n) and n that have been investigated in those works:…”