Square-free values of the Carmichael function

Abstract

We obtain an asymptotic formula for the number of square-free values among p À 1; for primes ppx; and we apply it to derive the following asymptotic formula for LðxÞ; the number of square-free values of the Carmichael function lðnÞ for 1pnpx; LðxÞ ¼ ðk þ oð1ÞÞ x ln 1Àa x ; where a ¼ 0:37395y is the Artin constant, and k ¼ 0:80328y is another absolute constant.

Divisibility of reduction in groups of rational numbers

“…This formula is consistent with the formula in [16,Theorem 1.3]. (5) An immediate consequence of the previous remark is that Γ,m = 0 for any group Γ and for any m. In fact, a ,m > 0 for any a ∈ Q * and if Γ ⊂ Q * is a subgroup with Γ ⊂ Γ, then ord p Γ | ord p Γ for any prime p ∈ Supp Γ.…”

“…For the second term observe that the Rankin Method (see [16,Lemma 3.3]) implies that for any c ∈ (0, 1), uniformly in m,…”

Roughly squarefree values of the Euler and Carmichael functions

“…In this section, we follow closely ideas from [9] that were used to establish (2). Our main result is the following analogue of Theorem 1 for the function λ:…”

“…To complete the proof, we can apply an analogue of Lemma 4 of [9] to deduce that the estimate holds for some absolute constant η(y) > 0. Taking κ(y) = η(y) e γα(y) Γ (α(y)) , we finish the proof.…”

“…Hence, the problem of estimating the number of integers n ≤ x for which ϕ(n) is squarefree reduces to that of estimating the number of primes p ≤ x for which p − 1 is squarefree. These questions have been previously investigated in [9], where it is shown that for any constant A > 0, the asymptotic relation (2) #{p ≤ x : p − 1 is squarefree} = α π(x) + O x log A x holds (see also [8]), and consequently, (3) #{n ≤ x : ϕ(n) is squarefree} = 3α 2 π(x) + O x log A x .…”