Certain Aspects of Some Arithmetic Functions in Number Theory

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Certain Aspects of Some Arithmetic Functions in Number Theory

¹

Nicu¸sor Minculete, Petric˘a Dicu

Abstract

The purpose of this paper is to present several inequalities about the arithmetic functions σ⁽ê⁾, τ⁽ê⁾, σ⁽ê⁾^∗, τ⁽ê⁾^∗ and other well-known arithmetic functions. Among these, we have the following:

pσ^∗k(n)·σl^∗(n) σ^∗k−l

2

(n) ≤ n^l⁻⁴^k ·σ^∗_k(n) +n^k⁻⁴^l·σ_l^∗(n) 2·σ^k−l

2 (n) ≤n

l−k

4 ·n^k+l² + 1

2 ,

for anyn, k, l∈N^∗, q

σ_k⁽ê⁾^∗(n)·τ⁽ê⁾^∗(n) σ⁽kê−⁾^∗l

2

(n) ≤n^l⁻⁴^k ·σ⁽kê^)∗(n) +n^k⁻⁴^l ·τ⁽ê⁾^∗(n) 2·σ⁽kê−⁾^∗l

2

(n) ≤

≤ n

l−k

4 · n^k+l² + 1

2 , for any n, k, l ∈ N^∗, σ_k⁽ê⁾(n)·σ_l⁽ê⁾(n) ≤ τ⁽ê⁾(n)· σ_k⁽ê₊⁾_l(n), for any n, k, l ∈ N^∗ and σ⁽_kê₊₁⁾^∗(n)

σ⁽_k^e⁾^∗(n) ≥ σ⁽^e⁾^∗(n)

τ⁽^e⁾^∗(n) ≥ τ(n), for any n, k∈N^∗, whereτ(n) is the number of the natural divisors ofnandσ(n) is the sum of the divisors ofn.

2000 Mathematics Subject Classification: 11A25

Key words and phrases: the sum of the natural divisors ofn, 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 ofn, 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 ofn.

1Received 3 November, 2009

Accepted for publication (in revised form) 16 June, 2010

135

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1 Introduction

Let n be a positive integer, n ≥ 1. We note with σk(n) the sum of the kth powers of divisors of n, so, σ^k(n) = X

d|n

d^k, whence we obtain the following equalities: σ₁(n) =σ(n) and σ₀(n) = τ(n)- the number of divisors of n (see [6]). If dis a unitary divisor ofn, then we have

d,n d

= 1. Letσk^∗(n) denote 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].

Letn >1 be an integer of canonical from n=pâ₁¹pâ₂²...pâr^r. The integer d =

r

Y

i=1

p^biⁱ is called an exponential divisor (or e-divisor) of n =

r

Y

i=1

pâiⁱ >1, if bⁱ|aⁱ for every i= 1, r. We note d|(e)n. Let σ⁽ê⁾(n) denote the sum of the exponential divisors of n and τ⁽ê⁾(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 d =

r

Y

i=1

p^biⁱ is called a e-unitary divisor of n =

r

Y

i=1

p^aiⁱ > 1 if bⁱ is a unitary divisor of ai, so

bi,ai

bi

= 1, for every i = 1, r. Let σ⁽ê⁾^∗(n) denote the sum of e-unitary divisor ofn, andτ⁽ê⁾^∗(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 ofn=pâ₁¹pâ₂²...pâr^r >1 isp₁p₂...p^r, where p₁p₂...pr=γ(n) is called the ”core” ofn.

Other aspects of these arithmetic function can be found in the papers [7] and [10].

In [6], J. S´andor shows that

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pσk(n)·σl(n) σ^k−l

2 (n) ≤n⁻⁽

k−l)

4 ·n

k+l

2 + 1

2 , for alln, k, l∈N^∗. In [8], J. S´andor and L. T´oth proved the inequalities

(2) n^k+ 1

2 ≥ σ^∗k(n) τ^∗(n) ≥√

n^k,

(3)

and

(3) σk^∗+m

σm^∗(n) ≥√ n^k, for all n≥1 and k, m≥0, real numbers.

In [3, 4], we found the inequalities (4)

pσ^k(n)·σ^l(n) σ^k−l

2 (n) ≤ n^l⁻⁴^kσk(n) +n^k⁻⁴^lσl(n) 2σ^k−l

2 (n) ≤n⁻⁽

k−l)

4 ·n^k⁺²^l + 1

2 ,

for everyn, k, l∈Nwithn≥1 and k−l 2 ∈N, (5)

pσk+2(n)·σk(n)

σ(n) ≤

√1nσk+2(n) +√nσk(n)

2σ(n) ≤ 1

√n·n^k⁺¹+ 1

2 ,

for everyn, k∈N andn≥1,

(6) q

σ_k⁽^e⁾(n)τ⁽^e⁾(n) σ^k−l

2 (n) ≤ n

l−k

4 σk⁽^e⁾(n) +n

k−l 4 τ⁽^e⁾(n) 2σ⁽k^e−⁾l

2

(n) ≤n⁻⁽

k−l)

4 ·n^k⁺²^l + 1

2 ,

for everyn, k, l∈Nwithn≥1 and k−l 2 ∈N,

(7) q

σ_k⁽ê₊₂⁾ (n)·τ⁽ê⁾(n) σ⁽ê⁾(n) ≤

√1nσk⁽^e+2⁾ (n) +√

nτ⁽^e⁾(n) 2σ⁽^e⁾(n) ≤ 1

√n ·n^k⁺¹+ 1

2 ,

for everyn, k∈N n≥1,

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q

σk⁽^e⁾(n)·τ⁽^e⁾(n)

τ⁽ê⁾(n) ≤ σ⁽kê⁾(n) +τ⁽ê⁾(n)

2τ⁽^e⁾(n) ≤ n^k+ 1 2 , and

(9) σk⁽^e⁾(n)

τ⁽^e⁾ ≤

n^k+ 1 2

2

, for everyn, k∈N andn≥1.

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2 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 for every b1, b2, ..., bⁿ ∈ R^∗ with m ≤ aⁱ

bⁱ ≤ M and m, M ∈ R, we have the following inequality:

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n

X

i=1

a²i +mM

n

X

i=1

b²i ≤(m+M)

n

X

i=1

aⁱbⁱ.

Theorem 1 For every n, k, l ∈ N with n ≥ 1 and k−l

2 ∈ N, the following relation

(11)

pσk^∗(n)·σ^∗l(n) σ^∗k−l

2

(n) ≤ n^l⁻⁴^k ·σk^∗(n) +n^k⁻⁴^l·σl^∗(n) 2·σ^∗k−l

2

(n) ≤n

l−k 4 ·n

k+l

2 + 1

2 is true.

Proof. Forn= 1, we have equality in relation (11). Forn≥2, in the Lemma above, making the substitutions aⁱ =

q

d^ki and bⁱ = 1 q

d^li

, where dⁱ is the unitary divisors of n, for all i= 1, τ^∗(n). Since 1≤ aⁱ

bⁱ = q

d^ki⁺^l ≤n

k+l

2 and

aibi=d

k−l 2

i , we takem= 1 andM =n^k⁺²^l. Therefore, inequality (10) becomes

τ∗(n)

X

i=1

d^ki +n

k+l 2 ·

τ∗(n)

X

i=1

1 d^li

≤ 1 +n

k+l 2

τ∗(n)

X

i=1

d

k−l 2

i , which is equivalent to

σk^∗(n) +n

k+l

2 ·σ^∗l(n) n^l ≤

1 +n

k+l 2

·σ^∗^k−l 2

(n), so that

(12) σ^∗k(n) +n

k−l

2 ·σ^∗l(n)≤ 1 +n

k+l 2

·σ^∗^k−l 2

(n), for everyn, k, l∈Nwithn≥2.

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The arithmetical mean is greater than the geometrical mean or they are equal, so for everyn, k, l∈Nwithn≥2, we have

(13)

q

n^k⁻²^l·σ^∗_k(n)·σ_l^∗(n)≤ σk^∗(n) +n

k−l

2 ·σl^∗(n)

2 .

Consequently, from the relations (12) and (13) and taking into account that the relation ”≤” is transitive, we deduce the inequality

pσ^∗k(n)·σ^∗l(n) σ^∗k−l

2

(n) ≤ n

l−k

4 ·σk^∗(n) +n

k−l

4 ·σl^∗(n) 2·σ^∗k−l

2

(n) ≤n

l−k 4 ·n

k+l

2 + 1

2 .

Remark 1 For k= l in inequality (11), we obtain the relation of J. S´andor and L. T´oth, namely

(14) n^k+ 1

2 ≥ σ_k^∗(n) τ^∗(n), for everyn, k∈N with n≥1.

Theorem 2 For every n, k, l ∈ N with n ≥ 1 and k−l

2 ∈ N, the following relation

(15) q

σk⁽ê⁾^∗(n)·τ⁽ê⁾^∗(n) σ⁽kê−⁾^∗l

2

(n) ≤ n^l⁻⁴^k ·σk⁽ê⁾^∗(n) +n^k⁻⁴^l ·τ⁽ê⁾^∗(n) 2·σ⁽kê−⁾^∗l

2

(n) ≤n

l−k 4 ·n

k+l

2 + 1

2 is true.

Proof. Forn= 1, we have equality in relation (15). Forn≥2, in the Lemma above, making the substitutions ai = q

d^ki and bi = 1 q

d^li

, where di is the e-unitary divisor of n, for all i= 1, τ⁽^e⁾^∗(n). Since k−l

2 ∈N, we havek≥l, so, we deduce 1≤ ai

bⁱ = q

d^ki⁺^l ≤n^k⁺²^l andaibi=d

k−l 2

i . Hence, we takem= 1 and M =n^k⁺²^l.

Therefore, inequality (10) becomes

τ(e)∗(n)

X

i=1

d^ki +n

k+l 2 ·

τ(e)∗(n)

X

i=1

1 d^li

≤ 1 +n

k+l 2

τ(e)∗(n)

X

i=1

d

k−l 2

i ,

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which is equivalent to

σ⁽^e⁾^∗(n) +n^k⁺²^l ·

τ(e)∗(n)

X

i=1

1 d^li

≤

1 +n^k⁺²^l σ⁽k^e−⁾^∗l

2

. But

τ⁽^e^)∗(n)

X

i=1

1 d^li

≥

τ⁽^e^)∗(n)

X

i=1

1

n^l = τ⁽^e⁾^∗(n) n^l . Therefore, we obtain the inequality

σ⁽k^e⁾^∗(n) +n

k+l

2 ·τ⁽^e⁾^∗(n) n^l ≤

1 +n

k+l 2

·σ⁽k^e−⁾^∗l 2

(n) which means that

(16) σ_k⁽^e⁾^∗(n) +n^k⁻²^l ·τ⁽^e⁾^∗(n)≤

1 +n^k⁺²^l

·σ⁽k^e−⁾^∗l 2

(n), for everyn, k, l∈Nwithn≥2.

The arithmetical mean is greater than the geometrical mean or they are equal, so for everyn, k, l ∈Nwithn≥2, we have

(17)

q

n^k⁻²^l ·σ_k⁽ê⁾^∗(n)·τ⁽ê⁾^∗(n)≤ σ⁽kê⁾^∗(n) +n

k−l

2 ·τ⁽^e⁾^∗(n)

2 .

Consequently, from the relations (16) and (17), we deduce the inequality q

σk⁽ê⁾^∗(n)·τ⁽ê⁾^∗(n) σ⁽kê−⁾^∗l

2

(n) ≤ n^l⁻⁴^k ·σ_k⁽ê⁾^∗(n) +n^k⁻⁴^l·τ⁽ê⁾^∗(n) 2·σ⁽kê−⁾^∗l

2

(n) ≤n

l−k

4 ·n^k⁺²^l + 1

2 .

Remark 2 For k=l, we obtain the relation

(18) σ_k⁽^e⁾^∗(n)

τ⁽^e⁾^∗(n) ≤

n^k+ 1 2

2

, for everyn, k∈Nwith n≥1.

Remark 3 For k=l= 1, we obtain the relation (19)

s

σ⁽^e⁾^∗(n)

τ⁽ê⁾^∗(n) ≤ σ⁽ê⁾^∗(n) +τ⁽ê⁾^∗(n)

2·τ⁽^e⁾^∗(n) ≤ n+ 1 2 for everyn, k∈Nwith n≥1.

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Remark 4 From inequality(19), we deduce another simple inequality, namely

(20) σ⁽^e⁾^∗(n)

τ⁽^e⁾^∗(n) ≤n for everyn≥1.

Theorem 3 For everyn, k, l∈Nwithn≥1, there are the following relations:

σ⁽_kê⁾(n)·σ_l⁽ê⁾(n)≤τ⁽ê⁾(n)·σ_k⁽ê₊⁾_l(n), (21)

σk⁽^e⁾(n)

σl⁽ê⁾(n) ≥ σ⁽ê⁾(n) τ⁽ê⁾(n)

!^k₋^l

≥τ^k⁻^l(n), (22)

σk⁽^e+1⁾ (n)

σ⁽k^e⁾(n) ≥τ(n) (23)

and

(24) σ_k⁽^e₊₁⁾ (n)

σk⁽^e⁾(n) ≥ σ⁽^e⁾(n)

τ⁽^e⁾(n) ≥τ(n)

Proof. For n= 1, we obtain equality in the relation above.

Letn=pâ₁¹pâ₂²...pâr^r >1. We apply Chebyshev’s Inequality for oriented system and, we deduce the inequality

σ⁽_k^e⁾(n)·σ_l⁽^e⁾(n) = X

d|(e)n

d^k· X

d|(e)n

d^l≤τ⁽^e⁾(n) X

d|(e)n

d^k⁺^l=τ⁽^e⁾(n)σ⁽_k^e₊⁾_l, so

σk⁽ê⁾(n)·σ⁽lê⁾(n)≤τ⁽ê⁾(n)·σ⁽kê+⁾l(n).

From [1], we shall use the inequality a^k₁+a^k₂ +...+a^kn

a^l₁+a^l₂+...+a^ln

≥

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 ofn, we obtain the following inequality:

X

d|(e)n

d^k X

d|(e)n

d^l ≥





 X

d|(e)n

d τ⁽^e⁾(n)







k−l

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which is equivalent to

σ⁽k^e⁾(n)

σ⁽lê⁾(n) ≥ σ⁽ê⁾(n) τ⁽ê⁾(n)

!^k₋^l .

We know from [5] that σ⁽^e⁾(n)

τ⁽^e⁾(n) ≥τ(n) and from the inequality σ_k⁽^e⁾(n)

σl⁽ê⁾(n) ≥ σ⁽ê⁾(n) τ⁽ê⁾(n)

!^k₋^l

, we deduce an interesting inequality, namely σ⁽k^e⁾(n)

σ⁽l^e⁾(n) ≥τ^k⁻^l(n).

We observe that making the substitution k→k+ 1 and l→k in inequality σ⁽k^e⁾(n)

σ⁽lê⁾(n) ≥ σ⁽ê⁾(n) τ⁽ê⁾(n)

!^k₋^l , we have

σk⁽^e+1⁾ (n)

σ_k⁽^e⁾(n) ≥τ(n).

If we assign values ofk from 1 tok−1, we have the following relations:

σ⁽kê⁾(n) ≥ τ(n)σ⁽kê−⁾1(n), σ⁽kê−⁾1(n) ≥ τ(n)σ⁽kê−⁾2(n),

...

σ⁽₂^e⁾(n) ≥ τ(n)σ⁽₁^e⁾(n),

and taking the product of these relations, we deduce the inequality σk⁽ê⁾(n)≥τ^k⁻¹(n)σ⁽ê⁾(n)≥τ^k(n)τ⁽ê⁾(n).

Therefore, we obtain

σ⁽_k^e⁾(n)≥τ^k(n)τ⁽^e⁾(n).

In relation σ⁽k^e⁾(n)

σ⁽_lê⁾(n) ≥ σ⁽ê⁾(n) τ⁽ê⁾(n)

!^k₋^l

, making the substitutions k→k+ 1 and l→k, we obtain the inequality

σ⁽_k^e₊₁⁾ (n)

σk⁽^e⁾(n) ≥ σ⁽^e⁾(n)

τ⁽^e⁾(n) ≥τ(n).

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Theorem 4 For everyn, k, l∈Nwithn≥1, there are the following relations:

σk⁽ê⁾^∗(n)·σ⁽lê⁾^∗(n)≤τ⁽ê⁾(n)·σ⁽kê+⁾^∗l(n), (25)

σk⁽^e⁾^∗(n)

σl⁽ê⁾^∗(n) ≥ σ⁽ê⁾^∗(n) τ⁽ê⁾^∗(n)

!^k₋^l

≥τ^k⁻^l(n), (26)

σ_k⁽^e₊₁⁾^∗(n)

σk⁽^e⁾^∗(n) ≥τ(n) (27)

and

(28) σ⁽_k^e₊₁⁾^∗(n)

σ⁽_k^e⁾^∗(n) ≥ σ⁽^e⁾^∗(n)

τ⁽^e⁾^∗(n) ≥τ(n).

Proof. We make the same proof as in Theorem 3, by repacing the exponential divisors with the e-unitary divisors.

References

[1] V. B˘andil˘a, M. Lascu and L. Panaitopol, Inegalit˘at¸i, Editura GIL, Zal˘au, 1995.

[2] M. O. Drimbe, Inegalit˘at¸i. Idei si metode, Editura GIL, Zal˘au, 2003.

[3] N. Minculete, Considerations concerning some inequalities of the arithmetic functions σk⁽ê⁾ and τ⁽ê⁾, A XII-a Conferint¸˘a Anual˘a a Societ˘at¸ii de Stiint¸e Matematice din România, Bac˘au, 2008.

[4] N. Minculete, Improvement of one of S´andor’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. S´andor, On Jordan’s Arithmetical Function, Gazeta Matematic˘a nr.

2-3/1993.

[7] J. S´andor, A Note on Exponential Divisors and Related Arithmetic Func- tions, Scientia Magna, Vol.1 (2006), No. 1.

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Quart.28 (1990), 255-258.

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[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. T´oth, On Certain Arithmetic Functions Involving Exponential Divi- sors, Annales Univ. Sci. Budapest., Sect. Comp. 24(2004), 285-294.

[11] L. T´oth and N. Minculete, Exponential unitary divisors (to appear in Annales Univ. Sci. Budapest., Sect. Comp.).

Nicu¸sor Minculete

”Dimitrie Catemir” University of Bra¸sov Str. Bisericii Romˆane, no. 107

Bra¸sov, Romˆania

e-mail: [email protected] Petric˘a Dicu

”Lucian Blaga” University of Sibiu Str. Dr. I. Rat¸iu, no. 5-7

Sibiu, Romˆania

e-mail: [email protected]