General Mathematics Vol. 19, No. 1 (2011), 135–144
Certain Aspects of Some Arithmetic Functions in
Number Theory 1
Nicuşor Minculete, Petrică Dicu
Abstract
The purpose of this paper is to present several inequalities about
the arithmetic functions σ (e) , τ (e) , σ (e)∗ , τ (e)∗ and other well-known
arithmetic functions. Among these, we have the following:
p ∗
k+l
l−k
k−l
l−k
σk (n) · σl∗ (n)
n 2 +1
n 4 · σk∗ (n) + n 4 · σl∗ (n)
4
·
≤
,
≤
n
σ ∗k−l (n)
2 · σ k−l (n)
2
2
2
for any n, k, l ∈ N∗ ,
q
(e)∗
σk (n) · τ (e)∗ (n)
(e)∗
≤
n
l−k
4
(e)∗
· σk
(n) + n
l−k
4
·
n
2
+1
≤
2 · σ k−l (n)
2
≤ n
· τ (e)∗ (n)
(e)∗
σ k−l (n)
k+l
2
k−l
4
2
, for any n, k, l ∈ N∗ ,
(e)
(e)
σk (n) · σl (n) ≤ τ (e) (n) ·
(e)∗
σ (e)∗ (n)
≥ τ (n), for any
(e)∗
τ (e)∗ (n)
σk (n)
n, k ∈ N∗ , where τ (n) is the number of the natural divisors of n and σ(n)
is the sum of the divisors of n.
(e)
σk+l (n), for any n, k, l ∈ N∗ and
σk+1 (n)
≥
2000 Mathematics Subject Classification: 11A25
Key words and phrases: the sum of the natural divisors of n, the number
of the natural divisors of n, the sum of the kth powers of the unitary divisors
of n, the number of the unitary divisors of n, the sum of the exponential
divisors of n, the number of the exponential divisors of n, the sum of the
e-unitary divisors of n, and the number of the e-unitary divisors of n.
1
Received 3 November, 2009
Accepted for publication (in revised form) 16 June, 2010
135
136
1
Nicuşor Minculete, Petrică Dicu
Introduction
Let n be a positive integer, n ≥ 1. We
Xnote with σk (n) the sum of the kth
powers of divisors of n, so, σk (n) =
dk , whence we obtain the following
d|n
equalities: σ1 (n) = σ(n) and σ0 (n) = τ (n)- thenumber
of divisors of n (see
n
[6]). If d is a unitary divisor of n, then we have d,
= 1. Let σk∗ (n) denote
d
the sum of the kth powers of the unitary divisors of n. We note d||n.
Next we have to mention that the notion of ”exponential divisor” was introduced M. V. Subbarao in [9].
Let n > 1 be an integer of canonical from n = pa11 pa22 ...par r .
r
Y
The integer d =
pbi i is called an exponential divisor (or e-divisor) of n =
i=1
r
Y
i=1
pai i > 1, if bi |ai for every i = 1, r. We note d|(e) n. Let σ (e) (n) denote the
sum of the exponential divisors of n and τ (e) (n) denote the number of the
exponential divisors of n. In [11] L. Tóth and N. Minculete introduced the
notion of ” exponential unitary divisors” or ”e-unitary divisors”. The integer
r
r
Y
Y
bi
d =
pi is called a e-unitary divisor of n =
pai i > 1 if bi is a unitary
i=1
i=1
ai
= 1, for every i = 1, r. Let σ (e)∗ (n) denote the
divisor of ai , so bi ,
bi
sum of e-unitary divisor of n, and τ (e)∗ (n) denote the number of the e-unitary
divisors of n. We note d|(e)∗ . By convention, 1 is an e-unitary divisor of
n > 1, the smallest e-unitary divisor of n = pa11 pa22 ...par r > 1 is p1 p2 ...pr , where
p1 p2 ...pr = γ(n) is called the ”core” of n.
Other aspects of these arithmetic function can be found in the papers [7] and
[10].
In [6], J. Sándor shows that
(1)
p
k+l
−(k−l)
σk (n) · σl (n)
n 2 +1
4
·
, for all n, k, l ∈ N∗ .
≤n
σ k−l (n)
2
2
In [8], J. Sándor and L. Tóth proved the inequalities
(2)
σ ∗ (n) √ k
nk + 1
≥ k∗
≥ n ,
2
τ (n)
137
Certain Aspects of Some Arithmetic Functions in Number Theory
and
∗
√
σk+m
≥
nk ,
∗ (n)
σm
(3)
for all n ≥ 1 and k, m ≥ 0, real numbers.
In [3, 4], we found the inequalities
(4)
p
σk (n) · σl (n)
n
≤
σ k−l (n)
l−k
4
2
k−l
k+l
−(k−l)
σk (n) + n 4 σl (n)
n 2 +1
,
≤n 4 ·
2σ k−l (n)
2
2
k−l
∈ N,
2
√
√1 σk+2 (n) + nσk (n)
n
for every n, k, l ∈ N with n ≥ 1 and
(5)
p
σk+2 (n) · σk (n)
≤
σ(n)
2σ(n)
1 nk+1 + 1
≤√ ·
,
2
n
for every n, k ∈ N and n ≥ 1,
(6)
q
(e)
σk (n)τ (e) (n)
σ k−l (n)
2
≤
n
l−k
4
(e)
σk (n) + n
(7)
τ (e) (n)
(e)
2σ k−l (n)
≤n
−(k−l)
4
·
(e)
σk+2 (n) · τ (e) (n)
σ (e) (n)
≤
√
(e)
√1 σ
(n) + nτ (e) (n)
n k+2
2σ (e) (n)
q
(e)
σk (n) · τ (e) (n)
τ (e) (n)
(e)
≤
+1
2
1 nk+1 + 1
,
≤√ ·
2
n
σk (n) + τ (e) (n)
nk + 1
≤
,
2
2τ (e) (n)
and
(e)
(9)
k+l
2
k−l
∈ N,
2
for every n, k ∈ N n ≥ 1,
(8)
n
2
for every n, k, l ∈ N with n ≥ 1 and
q
k−l
4
σk (n)
≤
τ (e)
for every n, k ∈ N and n ≥ 1.
nk + 1
2
2
,
,
138
2
Nicuşor Minculete, Petrică Dicu
Main results
An inequality which is due to J.B. Diaz and F.T. Matcalf is proved in [2],
namely:
Lemma 1 Let n be a positive integer, n ≥ 2. For every a1 , a2 , ..., an ∈ R and
ai
for every b1 , b2 , ..., bn ∈ R∗ with m ≤
≤ M and m, M ∈ R, we have the
bi
following inequality:
n
X
(10)
a2i
+ mM
i=1
n
X
b2i
i=1
≤ (m + M )
n
X
ai bi .
i=1
k−l
∈ N, the following
Theorem 1 For every n, k, l ∈ N with n ≥ 1 and
2
relation
p ∗
k−l
l−k
k+l
σk (n) · σl∗ (n)
l−k
n 4 · σk∗ (n) + n 4 · σl∗ (n)
n 2 +1
≤
≤n 4 ·
(11)
σ ∗k−l (n)
2 · σ ∗k−l (n)
2
2
2
is true.
Proof. For n = 1, we have equality in relation
(11). For n ≥ 2, in the Lemma
q
1
k
above, making the substitutions ai = di and bi = q , where di is the
dli
q
k+l
ai
= dk+l
unitary divisors of n, for all i = 1, τ ∗ (n). Since 1 ≤
≤ n 2 and
i
bi
k−l
ai bi = di 2 , we take m = 1 and M = n
τ ∗ (n)
τ ∗ (n)
X
dki
+n
k+l
2
i=1
k+l
2
. Therefore, inequality (10) becomes
(n)
τX
X 1
k−l
k+l
2
·
di 2 ,
≤
1
+
n
l
di
i=1
i=1
∗
which is equivalent to
σk∗ (n) + n
k+l
2
σk∗ (n) + n
k−l
2
so that
(12)
·
k+l
σl∗ (n) 2
· σ ∗k−l (n),
≤
1
+
n
nl
2
k+l
· σl∗ (n) ≤ 1 + n 2 · σ ∗k−l (n),
2
for every n, k, l ∈ N with n ≥ 2.
139
Certain Aspects of Some Arithmetic Functions in Number Theory
The arithmetical mean is greater than the geometrical mean or they are
equal, so for every n, k, l ∈ N with n ≥ 2, we have
k−l
q
k−l
σk∗ (n) + n 2 · σl∗ (n)
∗
∗
.
n 2 · σk (n) · σl (n) ≤
2
(13)
Consequently, from the relations (12) and (13) and taking into account that
the relation ”≤” is transitive, we deduce the inequality
p ∗
k−l
l−k
k+l
σk (n) · σl∗ (n)
l−k
n 4 · σk∗ (n) + n 4 · σl∗ (n)
n 2 +1
.
≤
≤n 4 ·
σ ∗k−l (n)
2 · σ ∗k−l (n)
2
2
2
Remark 1 For k = l in inequality (11), we obtain the relation of J. Sándor
and L. Tóth, namely
σ ∗ (n)
nk + 1
≥ k∗
,
2
τ (n)
(14)
for every n, k ∈ N with n ≥ 1.
Theorem 2 For every n, k, l ∈ N with n ≥ 1 and
relation
(15)
q
(e)∗
σk
(n) · τ (e)∗ (n)
(e)∗
σ k−l (n)
≤
n
l−k
4
(e)∗
· σk
2
(n) + n
(e)∗
k−l
4
2 · σ k−l (n)
k−l
∈ N, the following
2
· τ (e)∗ (n)
≤n
l−k
4
·
n
k+l
2
+1
2
2
is true.
Proof. For n = 1, we have equality in relation
(15). For n ≥ 2, in the Lemma
q
1
above, making the substitutions ai = dki and bi = q , where di is the
dli
k−l
∈ N, we have k ≥ l,
e-unitary divisor of n, for all i = 1, τ (e)∗ (n). Since
2
q
k−l
k+l
ai
so, we deduce 1 ≤
= dk+l
≤ n 2 and ai bi = di 2 . Hence, we take m = 1
i
bi
k+l
and M = n 2 .
Therefore, inequality (10) becomes
τ (e)∗ (n)
τ (e)∗ (n)
X
i=1
dki
+n
k+l
2
·
X
i=1
(e)∗
τ X(n) k−l
k+l
1
2
di 2 ,
≤
1
+
n
dli
i=1
140
Nicuşor Minculete, Petrică Dicu
which is equivalent to
τ (e)∗ (n)
σ
(e)∗
But
(n) + n
k+l
2
τ (e)∗ (n)
X
i=1
·
X
i=1
k+l
1
(e)∗
2
≤
1
+
n
σ k−l .
dli
2
(e)∗
τ
X(n) 1
1
τ (e)∗ (n)
=
.
≥
nl
nl
dli
i=1
Therefore, we obtain the inequality
+n
k+l
2
(n) + n
k−l
2
(e)∗
σk (n)
which means that
(e)∗
(16)
σk
k+l
τ (e)∗ (n) (e)∗
2
· σ k−l (n)
≤ 1+n
·
nl
2
k+l
(e)∗
· τ (e)∗ (n) ≤ 1 + n 2 · σ k−l (n),
2
for every n, k, l ∈ N with n ≥ 2.
The arithmetical mean is greater than the geometrical mean or they are equal,
so for every n, k, l ∈ N with n ≥ 2, we have
k−l
q
(e)∗
σk (n) + n 2 · τ (e)∗ (n)
k−l
(e)∗
(e)∗
2
· σk (n) · τ
(n) ≤
n
.
2
(17)
Consequently, from the relations (16) and (17), we deduce the inequality
q
k−l
l−k
(e)∗
k+l
(e)∗
σk (n) · τ (e)∗ (n)
l−k
n 4 · σk (n) + n 4 · τ (e)∗ (n)
n 2 +1
4 ·
.
≤
≤
n
(e)∗
(e)∗
2
σ k−l (n)
2 · σ k−l (n)
2
2
Remark 2 For k = l, we obtain the relation
k
2
(e)∗
σk (n)
n +1
≤
,
(18)
2
τ (e)∗ (n)
for every n, k ∈ N with n ≥ 1.
Remark 3 For k = l = 1, we obtain the relation
s
σ (e)∗ (n)
σ (e)∗ (n) + τ (e)∗ (n)
n+1
(19)
≤
≤
2
τ (e)∗ (n)
2 · τ (e)∗ (n)
for every n, k ∈ N with n ≥ 1.
Certain Aspects of Some Arithmetic Functions in Number Theory
141
Remark 4 From inequality (19), we deduce another simple inequality, namely
σ (e)∗ (n)
≤n
τ (e)∗ (n)
(20)
for every n ≥ 1.
Theorem 3 For every n, k, l ∈ N with n ≥ 1, there are the following relations:
(21)
(22)
(e)
(e)
(e)
σk (n) · σl (n) ≤ τ (e) (n) · σk+l (n),
!k−l
(e)
σk (n)
σ (e) (n)
≥ τ k−l (n),
≥
(e) (n)
(e)
τ
σ (n)
l
(e)
σk+1 (n)
(23)
(e)
σk (n)
≥ τ (n)
and
(e)
σk+1 (n)
(24)
(e)
σk (n)
≥
σ (e) (n)
≥ τ (n)
τ (e) (n)
Proof. For n = 1, we obtain equality in the relation above.
Let n = pa11 pa22 ...par r > 1. We apply Chebyshev’s Inequality for oriented system
and, we deduce the inequality
X
X
X
(e)
(e)
(e)
σk (n) · σl (n) =
dk ·
dl ≤ τ (e) (n)
dk+l = τ (e) (n)σk+l ,
d|(e) n
so
(e)
d|(e) n
d|(e) n
(e)
(e)
σk (n) · σl (n) ≤ τ (e) (n) · σk+l (n).
From [1], we shall use the inequality
ak1 + ak2 + ... + akn
≥
al1 + al2 + ... + aln
a1 + a2 + ... + an
n
k−l
,
for every a1 , a2 , ..., an > 0 and for all k, l ∈ N with k ≥ l, and by replacing
a1 , a2 , ..., with the exponential divisors of n, we obtain the following inequality:
X
 X k−l
dk
d


d|(e) n
 d|(e) n 
X
≥  (e)

 τ (n) 
dl
d|(e) n
142
Nicuşor Minculete, Petrică Dicu
which is equivalent to
(e)
σk (n)
(e)
σl (n)
σ (e) (n)
τ (e) (n)
≥
!k−l
.
σ (e) (n)
≥ τ (n) and from the inequality
We know from [5] that (e)
τ (n)
!k−l
(e)
σk (n)
σ (e) (n)
, we deduce an interesting inequality, namely
≥
(e)
τ (e) (n)
σ (n)
l
(e)
σk (n)
≥ τ k−l (n).
(e)
σl (n)
We observe that making the substitution k → k + 1 and l → k in inequality
!k−l
(e)
σk (n)
σ (e) (n)
≥
,
(e)
τ (e) (n)
σ (n)
l
we have
(e)
σk+1 (n)
≥ τ (n).
(e)
σk (n)
If we assign values of k from 1 to k − 1, we have the following relations:
(e)
(e)
σk (n) ≥ τ (n)σk−1 (n),
(e)
(e)
σk−1 (n) ≥ τ (n)σk−2 (n),
...
(e)
σ2 (n)
(e)
≥ τ (n)σ1 (n),
and taking the product of these relations, we deduce the inequality
(e)
σk (n) ≥ τ k−1 (n)σ (e) (n) ≥ τ k (n)τ (e) (n).
Therefore, we obtain
(e)
σk (n) ≥ τ k (n)τ (e) (n).
!k−l
(e)
σk (n)
σ (e) (n)
In relation (e)
, making the substitutions k → k + 1 and
≥
τ (e) (n)
σl (n)
l → k, we obtain the inequality
(e)
σk+1 (n)
(e)
σk (n)
≥
σ (e) (n)
≥ τ (n).
τ (e) (n)
Certain Aspects of Some Arithmetic Functions in Number Theory
143
Theorem 4 For every n, k, l ∈ N with n ≥ 1, there are the following relations:
(25)
(e)∗
σk
(e)∗
(26)
σk
(e)∗
(n)
(e)∗
σl (n)
(e)∗
(n) ≤ τ (e) (n) · σk+l (n),
!k−l
σ (e)∗ (n)
≥ τ k−l (n),
τ (e)∗ (n)
(n) · σl
≥
(e)∗
σk+1 (n)
(27)
(e)∗
σk
(n)
≥ τ (n)
and
(e)∗
(28)
σk+1 (n)
(e)∗
σk
(n)
≥
σ (e)∗ (n)
≥ τ (n).
τ (e)∗ (n)
Proof. We make the same proof as in Theorem 3, by repacing the exponential
divisors with the e-unitary divisors.
References
[1] V. Băndilă, M. Lascu and L. Panaitopol, Inegalităţi, Editura GIL, Zalău,
1995.
[2] M. O. Drimbe, Inegalităţi. Idei si metode, Editura GIL, Zalău, 2003.
[3] N. Minculete, Considerations concerning some inequalities of the arith(e)
metic functions σk and τ (e) , A XII-a Conferinţă Anuală a Societăţii de
Stiinţe Matematice din România, Bacău, 2008.
[4] N. Minculete, Improvement of one of Sándor’s inequalities, Octogon
Mathematical Magazine, vol. 17, no. 1 (2009).
[5] N. Minculete, Concerning some inequalities about arithmetic functions
which use the exponential divisors (to appear).
[6] J. Sándor, On Jordan’s Arithmetical Function, Gazeta Matematică nr.
2-3/1993.
[7] J. Sándor, A Note on Exponential Divisors and Related Arithmetic Functions, Scientia Magna, Vol.1 (2006), No. 1.
[8] J. Sándor and L. Tóth, On certain number-theoretic inequalities, Fib.
Quart. 28 (1990), 255-258.
144
Nicuşor Minculete, Petrică Dicu
[9] M. V. Subbarao, On some arithmetic convolutions in The Theory
of Arithmetic Functions, Lecture Notes in Mathematics, New York,
Springer-Verlag, 1972.
[10] L. Tóth, On Certain Arithmetic Functions Involving Exponential Divisors, Annales Univ. Sci. Budapest., Sect. Comp. 24 (2004), 285-294.
[11] L. Tóth and N. Minculete, Exponential unitary divisors (to appear in
Annales Univ. Sci. Budapest., Sect. Comp.).
Nicuşor Minculete
”Dimitrie Catemir” University of Braşov
Str. Bisericii Române, no. 107
Braşov, România
e-mail: minculeten@yahoo.com
Petrică Dicu
”Lucian Blaga” University of Sibiu
Str. Dr. I. Raţiu, no. 5-7
Sibiu, România
e-mail: petrica.dicu@ulbsibiu.ro