Some Properties of Extended Euler’s Function and Extended Dedekind’s Function

: 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.

Let K be an algebraic number field of degree [K : Q] = n, where n ∈N, n ≥ 2. We denote by O K the ring of integers of the field K. We denote by Spec(O K ) the set of the prime ideals of the ring O K . It is known that O K is a Dedekind domain. Let I be an ideal of O K . It is known that Euler's function and Dedekind's function were extended to the set of the ideals of the ring of integers O K . We denote this set by J. Accordingly, the extended Euler's function and the extended Dedekind's function ϕ ext ; ψ ext : J→ N * . These functions have been introduced, while taking into account that Dedekind domains have the factorization theorem for ideals analogous with the Fundamental theorem of arithmetic. Applying the fundamental theorem of Dedekind rings, there exist positive integers r and α i , i = 1, r and the different prime ideals P 1 , P 2 ,..., P r in the ring O K such that This decomposition is unique, except the order of factors.
In [3], Miguel defined the extended Euler totient function type for a non-zero ideal of a Dedekind domain, because the factorization of ideals is unique. He extended Menon's identity to residually finite Dedekind domains (rings of finite norm property).
The extended Euler's function for an ideal I of the ring O K is defined, as follows: where, by N (I), we meant the norm of the ideal I.
Other extended arithmetic functions in algebraic number fields can be found in [8].
We now recall some properties of the norm of an ideal, properties that we will use in proving our results. Proposition 1. ( [1,7,9]). Let K be an algebraic number field. Then: for (∀) nonzero ideals I, J of the ring O K .

Proposition 2.
( [1,7,9]). Let K be an algebraic number field. If I is an ideal of ring O K , such that N (I) is a prime number, then I∈Spec(O K ) .
We now consider the Riemann zeta function The Dedekind zeta function of a number field K This function can also be written as a Eulerian product, as follows: . Now, we recall a result about the quadratic fields; we will use this result in proving our results.

Proposition 4.
( [1,4,7]). Let a quadratic field K = Q √ d , where d = 0, 1 is a square free integer and let O K be the ring of integers of the quadratic field K. Afterwards, we have:

Some Results about Euler's Function, Dedekind's Function and Generalized Dedekind's Function
In [11], Sándor and Atanassov proved an inequality with the arithmetic functions ϕ and ψ given by for any integer n > 1. The above inequality is an improvement of the inequality given by Kannan and Srikanth [12], thus: Alzer and Kwong proved [13] another improvement, thus, for all n ≥ 2, we have Next, we give two refinements of inequality (1).

Proposition 5.
For any integer n > 1, we have the following inequality: This implies the following inequality: In (2), we take a = ϕ(n) n and b = ψ(n) n and deduce It is easy to prove by mathematical induction over n∈N, n ≥ 2, that: Consequently, we proved the statement.
In this inequality we make the substitution x → 1 x , y → 1 y and obtain: If we take λ = x x+y , then, we have 1 − λ = y x+y and we deduce the following inequality: For x = ϕ(n) n and y = ψ(n) n , we deduce the first part of the inequality of the statement. Since ϕ(n) + ψ(n) ≥ 2n and ϕ(n) + ψ(n) ≥ 2 ϕ(n) · ψ(n), for all n ∈ N * , and then, we proved the inequality of the statement.

Proposition 7.
For any integers t ≥ 2 and n ≥ 2, we have the following inequality: Therefore, we deduce the statement.
In 1988, Sierpinski and Schinzel ( [16][17][18]) proved the following inequality involving Euler's function: where n is non prime. Now, we give a similar result, for extended Euler's function: Proof. Since I is not a prime ideal of the Dedekind O K , it results that (∃) P i ∈ Spec(O K ), such that P 2 i |I or (∃) P i , P j ∈Spec(O K ) , P i = P j , such that P i · P j |I and N (P i ) ≤ N P j . In both cases, it results that N (P i ) ≤ N (I). Moreover, if P∈ Spec(O K ) such that P|I, we remark that 1 − 1 N(P) ∈(0, 1) . Using these, we obtain that: Proposition 10. Let n be a positive integer, n ≥ 2 and let K be an algebraic number field of degree [K : Q] = n. Afterwards: Proof. Because I is not a prime ideal of the Dedekind O K , it results that (∃) P i ∈ Spec(O K ) such that P 2 i |I or (∃) P i , P j ∈Spec(O K ) , P i = P j such that P i · P j |I and N (P i ) ≤ N P j . So, we obtain that N (P i ) ≤ N (I). Therefore, we have that: In 1940, T. Popovici ([19]) found the following inequality about Euler's function: Now, we give a similar result, for extended Euler's function: Proposition 11. Let n be a positive integer, n ≥ 2 and let K be an algebraic number field of degree [K : Q] = n. Subsequently: Proof. Let I and J be two ideals of the domain O K Since O K is a Dedekind ring, according to the fundamental theorem of Dedeking rings, (∃!) r, s, g∈N * , the different ideals P 1 , P 2 ,..., P r ,P 1 , P 2 ,..., P s ,P " 1 , P " 2 ,..., P " g ∈Spec(O K ) and α 1 , α 2 ,..., α r , β 1 , β 2 ,...β s , e 1 , e 2 ,...,e g , e g+1 ,..., e 2g , ∈ N * such that I = P α 1 1 · P α 2 2 · ... · P α r r · P " 1 e 1 · P " 2 e 2 · . . . · P " g e g and J = P 1 Applying the definition of the extended Euler's function and Proposition 1, we have: We return now to Proposition 7. If we take t = 2 in Proposition 7, we find a result from [20]: Here, we generalize this result, for extended Euler's function and extended Dedekind's function and for the algebraic number field K = Q √ 5 . . such that Using the formulas of ϕ ext (I) and ψ ext (I), it results that: where ζ K is the Dedekind zeta function of the quadratic field K = Q √ 5 . From these, it results that: However, according to Proposition 3, we have From (4) and (5), we obtain that: for (∀) nonzero ideal I of the ring Z 1+ (1 + 1/N (P)) ≥ 2.

Proposition 14.
Let n be a positive integer, n ≥ 2 and let K be an algebraic number field of degree [K : Q] = n. Afterwards, we have the following inequality:  and y = ψ ext (I) N(I) , we obtain the first part of the inequality of the statement. But, in the proof of Proposition 13, we showed that ϕ ext (I) + ψ ext (I) ≥ 2N (I) and since ϕ ext (I) + ψ ext (I) ≥ 2 ϕ ext (I) · ψ ext (I), for all nonzero ideal I of the ring O K , with N (I) ≥ 2, it results the inequality of the statement.

Conclusions
In this paper, starting from some inequalities satisfied by Euler's totient function ϕ and Dedekind's function ψ, we proved that the extended Euler's function ϕ ext and the extended Dedekind's function ψ ext satisfy similar inequalities.