Abstract
In this paper, we find some properties of Euler’s function and Dedekind’s function. We also generalize these results, from an algebraic point of view, for extended Euler’s function and extended Dedekind’s function, in algebraic number fields. Additionally, some known inequalities involving Euler’s function and Dedekind’s function, we generalize them for extended Euler’s function and extended Dedekind’s function, working in a ring of integers of algebraic number fields.
Highlights
Introduction and PreliminariesLet Euler’s function φ : N∗ → N∗, φ (n) = | {k ∈ N∗ |k ≤ n, (k, n) = 1} |, (∀) n ∈ N∗ . φ is α sometimes called Euler’s totient function
In [11], Sándor and Atanassov proved an inequality with the arithmetic functions φ and ψ given by n φ(n)+ψ(n) φ(n) + ψ(n) φ(n)+ψ(n)
Since φ(n) + ψ(n) ≥ 2n and φ(n) + ψ(n) ≥ 2 φ(n) · ψ(n), for all n ∈ N∗, and we proved the inequality of the statement
Summary
We denote by Spec(OK ) the set of the prime ideals of the ring OK . It is known that Euler’s function and Dedekind’s function were extended to the set of the ideals of the ring of integers OK . The extended Euler’s function for an ideal I of the ring OK is defined, as follows: φext ( I ) = N ( I ) · 1 −
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