Journal of Inequalities in Pure and
Applied Mathematics
ON AN INEQUALITY RELATED TO THE LEGENDRE
TOTIENT FUNCTION
volume 3, issue 3, article 37,
2002.
PENTTI HAUKKANEN
Department of Mathematics, Statistics and Philosophy
FIN-33014 University of Tampere,
Finland.
EMail: mapehau@uta.fi
Received 06 February, 2002;
accepted 27 February, 2002.
Communicated by: J. Sándor
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
017-02
Quit
Abstract
Let ∆(x, n) = ϕ(x, n) − xϕ(n)/n, where ϕ(x, n) is the Legendre totient function
and ϕ(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.
2000 Mathematics Subject Classification: 11A25
Key words: Legendre totient function; Inequality; Regular convolution; Unitary convolution.
On an Inequality Related to the
Legendre Totient Function
Pentti Haukkanen
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Generalization of (1.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4
Unitary Analogue of (1.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
References
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 14
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
http://jipam.vu.edu.au
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
hxi
X
(1.1)
ϕ(x, n) =
µ(d)
,
d
d|n
where µ is the Möbius function. A direct consequence of (1.1) is that
(1.2)
ϕ(x, n) =
xϕ(n)
+ O(θ(n)),
n
where θ(n) denotes the number of square-free divisors of n with θ(1) = 1. This
gives rise to the function ∆(x, n) defined as ∆(x, n) = ϕ(x, n) − xϕ(n)
. Suryan
narayana [8] obtains two inequalities 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 if x ≥ 1, n ≥ 2 and m = (n, [x]), then
ϕ(n)
1
1 θ(n) θ(m) mθ(m)ψ(n)
∆(x, n) + {x}
(1.3)
−
≤
+
−
,
n
2 m 2
2
nψ(m)
where {x} = x − [x] and ψ 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].
On an Inequality Related to the
Legendre Totient Function
Pentti Haukkanen
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 14
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
http://jipam.vu.edu.au
2.
Preliminaries
For each n let A(n) be a subset of the set of positive divisors of n. The elements of A(n) are said to be the A-divisors of n. The A-convolution of two
arithmetical functions f and g is defined by
n
X
(f ∗A g)(n) =
f (d)g
.
d
d∈A(n)
Narkiewicz [5] (see also [3]) defines an A-convolution to be regular if
(a) the set of arithmetical functions forms a commutative ring with unity with
respect to the ordinary addition and the A-convolution,
On an Inequality Related to the
Legendre Totient Function
Pentti Haukkanen
(b) the A-convolution of multiplicative functions is multiplicative,
Title Page
(c) the constant function 1 has an inverse µA with respect to the A-convolution,
and µA (n) = 0 or −1 whenever n is a prime power.
Contents
It can be proved [5] that an A-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 power pa (> 1) there exists a divisor t = τA (pa ) of a such
that
A(pa ) = 1, pt , p2t , . . . , prt ,
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 14
where rt = a, and
A(pit ) = 1, pt , p2t , . . . , pit , 0 ≤ i < r.
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
http://jipam.vu.edu.au
The positive integer t = τA (pa ) in part (ii) is said to be the A-type of pa . A
positive integer n is said to be A-primitive if A(n) = {1, n}. The A-primitive
numbers are 1 and pt , where p runs through the primes and t runs through the
A-types of the prime powers pa with a ≥ 1.
For all n let D(n) denote the set of all positive divisors of n and let U (n)
denote the set of all unitary divisors of n, that is,
n
o
n
= 1 = {d > 0 : d k n}.
U (n) = d > 0 : d n, d,
d
The D-convolution is the classical Dirichlet convolution and the U -convolution
is the unitary convolution [1]. These convolutions are regular with τD (pa ) = 1
and τU (pa ) = a for all prime powers pa (> 1).
Let k be a positive integer. We denote Ak (n) = d > 0 : dk ∈ A nk .
It is known [6] that the Ak -convolution is regular whenever the A-convolution
is regular. The symbol (m, n)A,k denotes the greatest kth power divisor of m
which belongs to A(n). In particular, (m, n)D,1 is the usual greatest common
divisor (m, n) of m and n, and (m, n)U,1 , usually written as (m, n)∗ , is the
greatest unitary divisor of n which is a divisor of m.
Throughout the rest of the paper A will be an arbitaray but fixed regular
convolution and k is a positive integer.
The A-analogue of the Möbius function µA is the multiplicative function
given by
−1 if pa (> 1) is A-primitive,
a
µA (p ) =
0 if pa is non-A-primitive.
In particular, µD = µ, the classical Möbius function, and µU = µ∗ , the unitary
analogue of the Möbius function [1].
On an Inequality Related to the
Legendre Totient Function
Pentti Haukkanen
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 14
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
http://jipam.vu.edu.au
The generalized Legendre totient function ϕA,k (x, n) is defined as the number of positive integers a ≤ x such that (a, nk )A,k = 1. It is known [2] that
(2.1)
ϕA,k (x, n) =
X
d∈Ak (n)
µAk (d)
hxi
dk
.
In particular, ϕA,k (n) = ϕA,k (nk , n). We recall that
Y
(2.2)
ϕA,k (n) = nk
1 − p−tk ,
On an Inequality Related to the
Legendre Totient Function
p|n
Q
where n = p pn(p) is the canonical factorization of n and t = τAk (pn(p) ), and
we define the generalized Dedekind totient function ψA,k as
Y
(2.3)
ψA,k (n) = nk
1 + p−tk .
p|n
Pentti Haukkanen
Title Page
Contents
If A is the Dirichlet convolution and k = 1, then ϕA,k (x, n), ϕA,k (n) and
ψA,k (n), respectively, reduce to the Legendre totient function, the Euler totient
function and the Dedekind totient function.
It follows from (2.1) that
x
X
ϕA,k (x, n) =
µAk (d) k + O(1)
d
d∈Ak (n)


X
xϕA,k (n)

+
O
µ2Ak (d)
=
k
n
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 14
d∈Ak (n)
=
xϕA,k (n)
+ O(θ(n)).
nk
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
http://jipam.vu.edu.au
This suggests we define
∆A,k (x, n) = ϕA,k (x, n) −
(2.4)
xϕA,k (n)
.
nk
We next present four lemmas which are needed in the proof of our inequality
for the function ∆A,k (x, n).
n o
Lemma 2.1. If f x, nk = n[x]k and mk = [x], nk A,k , then
(i) f x, n
(ii)
mk
nk
Proof.
k
On an Inequality Related to the
Legendre Totient Function
= 0 if m = n,
≤ f (x, nk ) ≤ 1 −
mk
nk
Pentti Haukkanen
if m < n.
(i) If m = n, then nk |[x] and thus
n
[x]
nk
o
= 0.
k
k
k
(ii) Let m < n.
n Then
o n 6 | [x] and thus [x] = an + r, where
n 0o< r < n .
Therefore n[x]k = nrk , where 0 < r < nk , that is, n1k ≤ n[x]k ≤ 1 − n1k .
Now, writing
[x]
nk
=
([x]/mk )
(nk /mk )
we arrive at our result.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Lemma 2.2. For n ≥ 2
X
d∈Ak (n)
ω(d) is odd
µ2Ak (d) =
θ(n)
,
2
Quit
Page 7 of 14
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
where ω(d) is the number of distinct prime divisors of d.
http://jipam.vu.edu.au
Proof. It is clear that
X
µ2Ak (d)
d∈Ak (n)
ω(d) is odd
=
ω(n)
ω(n)
θ(n)
+
+ · · · = 2ω(n)−1 =
.
1
3
2
Lemma 2.3. Let mk = ([x], nk )A,k . Then
On an Inequality Related to the
Legendre Totient Function
X µ2A (d)([x], dk )A,k
mk θ(m)ψA,k (n)
k
=
.
dk
nk ψA,k (m)
Pentti Haukkanen
d∈Ak (n)
Proof. By multiplicativity it is enough to consider the case in which n is a prime
power. For the sake of brevity we do not present the details.
Title Page
Contents
Lemma 2.4. We have
X
ϕA,k (n)
∆A,k (x, n) + {x}
=
−
µAk (d)f (x, dk ).
nk
d∈Ak (n)
JJ
J
II
I
Go Back
Proof. Clearly
Close
ϕA,k (x, n) =
X
µAk (d)
d∈Ak (n)
=
xϕA,k (n)
−
nk
x
dk
−
X
d∈Ak (n)
n x o
Quit
dk
µAk (d)
Page 8 of 14
nxo
dk
.
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
http://jipam.vu.edu.au
Thus
∆A,k (x, n) = −
X
µAk (d)
d∈Ak (n)
It can be verified that
x
dk
=
{x}
dk
+
nxo
dk
.
n o
[x]
. Thus
dk
X
ϕA,k (n)
[x]
∆A,k (x, n) = −{x}
−
µAk (d)
.
k
n
dk
d∈Ak (n)
This completes the proof.
On an Inequality Related to the
Legendre Totient Function
Pentti Haukkanen
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 14
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
http://jipam.vu.edu.au
3.
Generalization of (1.3)
Theorem 3.1. Let x ≥ 1, n ≥ 2 and mk = ([x], nk )A,k . Then
(3.1)
∆A,k (x, n) + {x} ϕA,k (n) − 1 1 nk
2 m θ(n) θ(m) mk θ(m)ψA,k (n)
+
−
.
≤
2
2
nk ψA,k (m)
On an Inequality Related to the
Legendre Totient Function
Proof. Firstly, suppose that m = n, that is, nk | [x]. Then
Pentti Haukkanen
X µA (d)[x]
ϕA,k (n)
k
ϕA,k (x, n) =
= [x]
.
k
d
nk
d∈Ak (n)
Title Page
Thus
{x}ϕA,k (n)
∆A,k (x, n) +
= 0.
nk
Since n ≥ 2, the left-hand side of (3.1) is = 0. Therefore (3.1) holds.
Secondly, suppose that 1 ≤ m < n. Then, by Lemma 2.4,
{x}ϕA,k (n)
∆A,k (x, n)+
=
nk
d∈Ak (n)
ω(d) is odd
µ2Ak (d)f (x, dk )−
X
d∈Ak (n)
ω(d) is even
µ2Ak (d)f (x, dk ).
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 14
By Lemma 2.1,
∆A,k (x, n) +
X
Contents
{x}ϕA,k (n)
nk
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
http://jipam.vu.edu.au
≤
X
µ2Ak (d)
d∈Ak (n)
ω(d) is odd
dk 6 |[x]
=
X
([x], dk )A,k
1−
dk
µ2Ak (d) −
d∈Ak (n)
ω(d) is odd
−
X
X
−
X
d∈Ak (n)
ω(d) is even
dk 6 |[x]
µ2Ak (d)
([x], dk )A,k
dk
µ2Ak (d)
d∈Ak (n)
ω(d) is odd
dk |[x]
µ2Ak (d)
d∈Ak (n)
X
([x], dk )A,k
([x], dk )A,k
2
+
µ
(d)
.
Ak
k
dk
d
d∈A (n)
k
dk |[x]
By Lemmas 2.2 and 2.3 and definition of the number m,
∆A,k (x, n)+
{x}ϕA,k (n)
θ(n)
≤
−
nk
2
X
d∈Ak (m)
ω(d) is odd
µ2Ak (d)−
mk θ(m)ψA,k (n)
+θ(m).
nk ψA,k (m)
We distinguish the cases m = 1 and m > 1 and apply Lemma 2.2 in the case
m > 1 to obtain
θ(n) θ(m) mk θ(m)ψA,k (n)
{x}ϕA,k (n) 1 1
−
≤
+
−
.
∆A,k (x, n) +
nk
2 m
2
2
nk ψA,k (m)
In a similar way we can show that
{x}ϕA,k (n) 1 1
θ(n) θ(m) mk θ(m)ψA,k (n)
∆A,k (x, n) +
−
≥
−
−
+
.
nk
2 m
2
2
nk ψA,k (m)
On an Inequality Related to the
Legendre Totient Function
Pentti Haukkanen
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 14
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
This completes the proof.
http://jipam.vu.edu.au
Remark 3.1. If A is the Dirichlet convolution and k = 1, then (3.1) reduces to
(1.3).
On an Inequality Related to the
Legendre Totient Function
Pentti Haukkanen
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 14
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
http://jipam.vu.edu.au
4.
Unitary Analogue of (1.3)
We recall that a positive integer d is said to be a unitary divisor of n (written as
dkn) if d is a divisor of n and d, nd = 1. The unitary analogue of the Legendre
totient function ϕ∗ (x, n) is the number of positive integers a ≤ x such that
(a, n)∗ = 1. Its arithmetical expression is
hxi
X
µ∗ (d)
ϕ∗ (x, n) =
.
d
dkn
In particular, the unitary analogue of the Euler totient function is given by
ϕ∗ (n) = ϕ∗ (n, n). We define the unitary analogue of the Dedekind totient
function as
Y
ψ ∗ (n) = n
(1 + p−e ).
pe kn
It is easy to see that ψ ∗ (n) = σ ∗ (n), where σ ∗ (n) is the sum of the unitary
∗
divisors of n. The function ∆∗ (x, n) is defined as ∆∗ (x, n) = ϕ∗ (x, n) − xϕn(n) .
The unitary analogue of (1.3) is
∗
∗
ϕ
(n)
1
1 θ(n) θ(m) mθ(m)σ ∗ (n)
(4.1) ∆ (x, n) + {x}
−
≤
+
−
,
n
2 m 2
2
nσ ∗ (m)
where m = ([x], n)∗ . In fact, if A is the unitary convolution and k = 1, then
(3.1) reduces to (4.1).
On an Inequality Related to the
Legendre Totient Function
Pentti Haukkanen
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 14
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
http://jipam.vu.edu.au
References
[1] E. COHEN, Arithmetical functions associated with the unitary divisors of
an integer, Math. Z. 74 (1960), 66–80.
[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. MITRINOVIĆ, J. SÁNDOR AND B. CRSTICI, Handbook of Number
Theory, Kluwer Academic Publishers, MIA Vol. 351, 1996.
[5] W. NARKIEWICZ, On a class of arithmetical convolutions, Colloq. Math.,
10 (1963), 81–94.
[6] V. SITA RAMAIAH, Arithmetical sums in regular convolutions, J. Reine
Angew. Math., 303/304 (1978), 265–283.
[7] A. SIVARAMASARMA, On ∆(x, n) = ϕ(x, n) − xϕ(n)/n, Math. Stud.,
46 (1978), 160–164.
[8] D. SURYANARAYANA, On ∆(x, n) = ϕ(x, n) − xϕ(n)/n, Proc. Amer.
Math. Soc., 44 (1974), 17–21.
On an Inequality Related to the
Legendre Totient Function
Pentti Haukkanen
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 14
J. Ineq. Pure and Appl. Math. 3(3) Art. 37, 2002
http://jipam.vu.edu.au