Let∆(x, n) =ϕ(x, n)−xϕ(n)/n, whereϕ(x, n)is the Legendre totient function andϕ(n)is the Euler totient function

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Volume 3, Issue 3, Article 37, 2002

ON AN INEQUALITY RELATED TO THE LEGENDRE TOTIENT FUNCTION

PENTTI HAUKKANEN

DEPARTMENT OFMATHEMATICS, STATISTICS ANDPHILOSOPHY

FIN-33014 UNIVERSITY OFTAMPERE, FINLAND.

[email protected]

Received 06 February, 2002; accepted 27 February, 2002 Communicated by J. Sándor

ABSTRACT. Let∆(x, n) =ϕ(x, n)−xϕ(n)/n, whereϕ(x, n)is the Legendre totient function andϕ(n)is the Euler totient function. An inequality for∆(x, n)is known. In this paper we give a unitary analogue of this inequality, and more generally we give this inequality in the setting of regular convolutions.

Key words and phrases: Legendre totient function; Inequality; Regular convolution; Unitary convolution.

2000 Mathematics Subject Classification. 11A25.

1. INTRODUCTION

The Legendre totient function ϕ(x, n) is defined as the number of positive integers ≤ x which are prime to n. The Euler totient function ϕ(n)is a special case of ϕ(x, n). Namely, ϕ(n) =ϕ(n, n). It is well known that

(1.1) ϕ(x, n) = X

d|n

µ(d)hx d i

,

whereµis the Möbius function. A direct consequence of (1.1) is that

(1.2) ϕ(x, n) = xϕ(n)

n +O(θ(n)),

whereθ(n)denotes the number of square-free divisors ofnwithθ(1) = 1. This gives rise to the function∆(x, n)defined as∆(x, n) = ϕ(x, n)−^xϕ(n)_n . Suryanarayana [8] obtains two inequal- ities for the function∆(x, n). Sivaramasarma [7] establishes an inequality which sharpens the first inequality and contains as a special case the second inequality of Suryanarayana [8]. The inequality of Sivaramasarma [7] states that ifx≥1,n≥2andm= (n,[x]), then

(1.3)

∆(x, n) +{x}ϕ(n) n − 1

2 1

m

≤ θ(n)

2 + θ(m)

2 − mθ(m)ψ(n) nψ(m) ,

ISSN (electronic): 1443-5756

c 2002 Victoria University. All rights reserved.

017-02

where{x}=x−[x]andψ is the Dedekind totient function. See also [4, §I.32].

In this paper we give (1.3) in the setting of Narkiewicz’s regular convolution and the kth power greatest common divisor. As special cases we obtain (1.3) and its unitary analogue. The proof is adapted from that given by Sivaramasarma [7].

2. PRELIMINARIES

For each n let A(n) be a subset of the set of positive divisors ofn. The elements ofA(n) are said to be theA-divisors ofn. TheA-convolution of two arithmetical functionsf andg is defined by

(f ∗Ag)(n) = X

d∈A(n)

f(d)g n

d

. Narkiewicz [5] (see also [3]) defines anA-convolution to be regular if

(a) the set of arithmetical functions forms a commutative ring with unity with respect to the ordinary addition and theA-convolution,

(b) theA-convolution of multiplicative functions is multiplicative,

(c) the constant function1has an inverseµ_Awith respect to theA-convolution, andµ_A(n) = 0or−1whenevernis a prime power.

It can be proved [5] that anA-convolution is regular if and only if (i) A(mn) = {de:d∈A(m), e∈A(n)}whenever(m, n) = 1,

(ii) for each prime powerp^a(>1)there exists a divisort =τ_A(p^a)ofasuch that A(p^a) =

1, p^t, p^2t, . . . , p^rt , wherert=a, and

A(p^it) =

1, p^t, p^2t, . . . , p^it ,0≤i < r.

The positive integert = τ_A(p^a)in part (ii) is said to be theA-type ofp^a. A positive integern is said to beA-primitive ifA(n) = {1, n}. TheA-primitive numbers are1andp^t, wherepruns through the primes andtruns through theA-types of the prime powersp^awitha≥1.

For allnletD(n)denote the set of all positive divisors ofnand letU(n)denote the set of all unitary divisors ofn, that is,

U(n) = n

d >0 :d n,

d,n d

= 1o

={d >0 :dkn}.

TheD-convolution is the classical Dirichlet convolution and the U-convolution is the unitary convolution [1]. These convolutions are regular withτ_D(p^a) = 1andτ_U(p^a) = afor all prime powersp^a(>1).

Letkbe a positive integer. We denoteA_k(n) =

d >0 :d^k ∈A n^k . It is known [6] that the Ak-convolution is regular whenever the A-convolution is regular. The symbol (m, n)A,k

denotes the greatestkth power divisor ofmwhich belongs toA(n). In particular,(m, n)D,1 is the usual greatest common divisor(m, n)ofmandn, and(m, n)_U,1, usually written as(m, n)^∗, is the greatest unitary divisor ofnwhich is a divisor ofm.

Throughout the rest of the paperAwill be an arbitaray but fixed regular convolution andkis a positive integer.

TheA-analogue of the Möbius functionµ_Ais the multiplicative function given by µ_A(p^a) =

−1 ifp^a(>1)isA-primitive, 0 ifp^ais non-A-primitive.

In particular,µ_D =µ, the classical Möbius function, andµ_U =µ^∗, the unitary analogue of the Möbius function [1].

The generalized Legendre totient function ϕ_A,k(x, n) is defined as the number of positive integersa≤xsuch that(a, n^k)A,k = 1. It is known [2] that

(2.1) ϕA,k(x, n) = X

d∈A_k(n)

µA_k(d) h x

d^k i

.

In particular,ϕ_A,k(n) =ϕ_A,k(n^k, n). We recall that

(2.2) ϕ_A,k(n) = n^kY

p|n

1−p^−tk , wheren = Q

pp^n(p) is the canonical factorization ofn and t = τ_Ak(p^n(p)), and we define the generalized Dedekind totient functionψA,k as

(2.3) ψ_A,k(n) = n^kY

p|n

1 +p^−tk .

IfAis the Dirichlet convolution andk = 1, thenϕ_A,k(x, n),ϕ_A,k(n)andψ_A,k(n), respectively, reduce to the Legendre totient function, the Euler totient function and the Dedekind totient function.

It follows from (2.1) that

ϕ_A,k(x, n) = X

d∈A_k(n)

µ_Ak(d)x

d^k +O(1)

= xϕ_A,k(n) n^k +O



 X

d∈A_k(n)

µ²_A

k(d)





= xϕ_A,k(n)

n^k +O(θ(n)).

This suggests we define

(2.4) ∆_A,k(x, n) =ϕ_A,k(x, n)− xϕ_A,k(n) n^k .

We next present four lemmas which are needed in the proof of our inequality for the function

∆_A,k(x, n).

Lemma 2.1. Iff x, n^k

=n

[x]

n^k

o

andm^k = [x], n^k

A,k, then (i) f x, n^k

= 0ifm=n,

(ii) ^m_nk^k ≤f(x, n^k)≤1− ^m_nk^k ifm < n.

Proof. (i) Ifm =n, thenn^k|[x]and thusn_[x]

n^k

o

= 0.

(ii) Let m < n. Then n^k 6 | [x] and thus [x] = an^k +r, where0 < r < n^k. Therefore n[x]

n^k

o

= _n^rk, where 0 < r < n^k, that is, _n¹k ≤ n

[x]

n^k

o ≤ 1 − _n¹k. Now, writing

[x]

n^k = ^([x]/m_(nk/m^kk)) we arrive at our result.

Lemma 2.2. Forn ≥2

X

d∈Ak(n) ω(d) is odd

µ²_A

k(d) = θ(n) 2 , whereω(d)is the number of distinct prime divisors ofd.

Proof. It is clear that X

d∈Ak(n) ω(d) is odd

µ²_A

k(d) =

ω(n) 1

+

ω(n) 3

+· · ·= 2^ω(n)−1 = θ(n) 2 .

Lemma 2.3. Letm^k = ([x], n^k)_A,k. Then

X

d∈Ak(n)

µ²_Ak(d)([x], d^k)A,k

d^k = m^kθ(m)ψ_A,k(n) n^kψA,k(m) .

Proof. By multiplicativity it is enough to consider the case in whichnis a prime power. For the

sake of brevity we do not present the details.

Lemma 2.4. We have

∆_A,k(x, n) +{x}ϕ_A,k(n)

n^k =− X

d∈A_k(n)

µ_Ak(d)f(x, d^k).

Proof. Clearly

ϕ_A,k(x, n) = X

d∈A_k(n)

µ_Ak(d)x

d^k −nx d^k

o

= xϕ_A,k(n)

n^k − X

d∈A_k(n)

µ_Ak(d)nx d^k

o .

Thus

∆_A,k(x, n) =− X

d∈A_k(n)

µ_Ak(d)nx d^k

o .

It can be verified that_x

d^k = ^{x}_dk + n[x]

d^k

o . Thus

∆_A,k(x, n) =−{x}ϕ_A,k(n)

n^k − X

d∈A_k(n)

µ_Ak(d) [x]

d^k

.

This completes the proof.

3. GENERALIZATION OF (1.3) Theorem 3.1. Letx≥1,n ≥2andm^k= ([x], n^k)_A,k. Then (3.1)

∆_A,k(x, n) +{x}ϕ_A,k(n) n^k − 1

2 1

m

≤ θ(n)

2 + θ(m)

2 − m^kθ(m)ψ_A,k(n) n^kψA,k(m) .

Proof. Firstly, suppose thatm=n, that is,n^k |[x]. Then ϕ_A,k(x, n) = X

d∈A_k(n)

µA_k(d)[x]

d^k = [x]ϕA,k(n) n^k . Thus

∆_A,k(x, n) + {x}ϕ_A,k(n) n^k = 0.

Sincen ≥2, the left-hand side of (3.1) is= 0. Therefore (3.1) holds.

Secondly, suppose that1≤m < n. Then, by Lemma 2.4,

∆_A,k(x, n) + {x}ϕ_A,k(n)

n^k = X

d∈Ak(n) ω(d) is odd

µ²_A

k(d)f(x, d^k)− X

d∈Ak(n) ω(d) is even

µ²_A

k(d)f(x, d^k).

By Lemma 2.1,

∆A,k(x, n) + {x}ϕ_A,k(n) n^k

≤ X

d∈Ak(n) ω(d) is odd

dk6 |[x]

µ²_Ak(d)

1− ([x], d^k)_A,k d^k

− X

d∈Ak(n) ω(d) is even

dk6 |[x]

µ²_Ak(d)([x], d^k)_A,k d^k

= X

d∈Ak(n) ω(d) is odd

µ²_Ak(d)− X

d∈Ak(n) ω(d) is odd

dk|[x]

µ²_Ak(d)

− X

d∈A_k(n)

µ²_Ak(d)([x], d^k)_A,k

d^k + X

d∈Ak(n) dk|[x]

µ²_Ak(d)([x], d^k)_A,k d^k .

By Lemmas 2.2 and 2.3 and definition of the numberm,

∆_A,k(x, n) + {x}ϕ_A,k(n)

n^k ≤ θ(n)

2 − X

d∈Ak(m) ω(d) is odd

µ²_A

k(d)− m^kθ(m)ψ_A,k(n)

n^kψ_A,k(m) +θ(m).

We distinguish the casesm= 1andm >1and apply Lemma 2.2 in the casem >1to obtain

∆A,k(x, n) + {x}ϕA,k(n) n^k −1

2 1

m

≤ θ(n)

2 + θ(m)

2 − m^kθ(m)ψA,k(n) n^kψ_A,k(m) . In a similar way we can show that

∆_A,k(x, n) + {x}ϕ_A,k(n) n^k −1

2 1

m

≥ −θ(n)

2 − θ(m)

2 + m^kθ(m)ψ_A,k(n) n^kψA,k(m) .

This completes the proof.

Remark 3.2. IfAis the Dirichlet convolution andk = 1, then (3.1) reduces to (1.3).

4. UNITARYANALOGUE OF(1.3)

We recall that a positive integerdis said to be a unitary divisor ofn(written asdkn) ifdis a divisor ofn and d,ⁿ_d

= 1. The unitary analogue of the Legendre totient functionϕ^∗(x, n)is the number of positive integersa≤xsuch that(a, n)^∗ = 1. Its arithmetical expression is

ϕ^∗(x, n) = X

dkn

µ^∗(d)hx d i

.

In particular, the unitary analogue of the Euler totient function is given byϕ^∗(n) = ϕ^∗(n, n).

We define the unitary analogue of the Dedekind totient function as ψ^∗(n) = nY

p^ekn

(1 +p^−e).

It is easy to see thatψ^∗(n) = σ^∗(n), whereσ^∗(n)is the sum of the unitary divisors ofn. The function∆^∗(x, n)is defined as∆^∗(x, n) = ϕ^∗(x, n)− ^xϕ∗_n⁽ⁿ⁾.

The unitary analogue of (1.3) is (4.1)

∆^∗(x, n) +{x}ϕ^∗(n) n − 1

2 1

m

≤ θ(n)

2 +θ(m)

2 − mθ(m)σ^∗(n) nσ^∗(m) ,

wherem = ([x], n)^∗. In fact, ifA is the unitary convolution andk = 1, then (3.1) reduces to (4.1).

REFERENCES

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[2] P. HAUKKANEN, Some generalized totient functions, Math. Stud. 56 (1988), 65–74.

[3] P.J. McCARTHY, Introduction to Arithmetical Functions, Springer–Verlag, New York, 1986.

[4] D.S. MITRINOVI ´C, J. SÁNDOR AND B. CRSTICI, Handbook of Number Theory, Kluwer Aca- demic Publishers, MIA Vol. 351, 1996.

[5] W. NARKIEWICZ, On a class of arithmetical convolutions, Colloq. Math., 10 (1963), 81–94.

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[7] A. SIVARAMASARMA, On∆(x, n) =ϕ(x, n)−xϕ(n)/n, Math. Stud., 46 (1978), 160–164.

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