On the mean value of the Pseudo-Smarandache-Squarefree function

Scientia Magna Vol. 4 (2008), No. 1, 86-89

On the mean value of the Pseudo-Smarandache-Squarefree function Xuhui Fan†‡ and Chengliang Tian† † Department of Mathematics, Northwest University, Xi’an, Shaanxi, 710069 ‡ Foundation Department, Engineering College of Armed Police Force, Xi’an, Shaanxi, 710086 Abstract For any positive integer n, the Pseudo Smarandache Squarefree function Zw (n) n is defined as Zw (n) = min{m : n|m ½ ¾ , m ∈ N }, and the function Z(n) is defined as Z(n) = m(m + 1) , m ∈ N . The main purpose of this paper is using the elementary min m : n ≤ 2 methods to study the mean value properties of the function Zw (Z(n)), and give a sharper mean value formula for it.

Keywords Pseudo-Smarandache-Squarefree function Zw (n), function Z(n), mean value, asymptotic formula.

§1. Introduction and result For any positive integer n, the Pseudo-Smarandache-Squarefree function Zw (n) is defined as the smallest positive integer m such that n | mn . That is, Zw (n) = min{m : n|mn , m ∈ N }.

For example Zw (1) = 1, Zw (2) = 2, Zw (3) = 3, Zw (4) = 2, Zw (5) = 5, Zw (6) = 6, Zw (7) = 7, Zw (8) = 2, Zw (9) = 3, Zw (10) = 10, · · · . About the elementary properties of Zw (n), some authors had studied it, and obtained some interesting results. For example, Felice Russo [1] obtained some elementary properties of Zw (n) as follows: Property 1. The function Zw (n) is multiplicative. That is, if GCD(m, n) = 1, then Zw (m · n) = Zw (m) · Zw (n).

Property 2. Zw (n) = n if and only if n is a squarefree number.

The main purpose of this paper is using the elementary method to study the mean value properties of½Zw (Z(n)), and give a sharper ¾ asymptotic formula for it, where Z(n) is defined as m(m + 1) Z(n) = min m : n ≤ , m ∈ N . That is, we shall prove the following conclusion: 2 Theorem. For any real number x ≥ 2, we have the asymptotic formula ! √ Ã ´ ³ 5´ Y³ X 3 4 2 1 2 + O x4 · · x , Zw (Z(n)) = 1 + 1+ p(p2 − 1) π2 p n≤x

where

Y p

denotes the product over all primes.

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On the mean value of the Pseudo-Smarandache-Squarefree function

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§2. Some lemmas To complete the proof of the theorem, we need the following several lemmas.

Lemma 1. For any real number x ≥ 2, we have the asymptotic formula X √ 6 µ2 (m) = 2 x + O( x). π

(1)

m≤x

Proof. See reference [2].

Lemma 2. For any real number x ≥ 2, we have the asymptotic formula ³ 5´ X 2 m2 = 2 x3 + O x 2 , π m≤x m∈A

where A denotes the set of all square-free integers.

Proof. By the Abel’s summation formula (See Theorem 4.2 of [3]) and Lemma 1, we have X

m2

X

=

m2 µ2 (m) = x2 ·

m≤x

m≤x

m∈A

Z x ³ ³6 √ ´ √ ´ 6 x + O( x) − 2 t t + O( t) dt π2 π2 1

³ 5´ ³ 5´ 4 3 6 3 2 3 2 − x + O x x = x + O x2 . π2 π2 π2 This proves Lemma 2.

Lemma 3. For any real number x ≥ 2 and s > 1 , we have the inequality ´ X Zw (m) Y ³ 1 < 1 + s−1 s . s m p (p − 1) p m≤x =

m∈B

Specially, if s >

3 , then we have the asymptotic formula 2 ´ ³ 3 ´ X Zw (m) Y ³ 1 = 1 + + O x 2 −s , s s−1 (ps − 1) m p p m≤x m∈B

where B denotes the set of all square-full integers.

Proof. First we define the arithmetical function a(m) as follows:   1 if m ∈ B ; a(m) =  0 otherwise.

From Property 1 and the definition of a(m) we know that the function Zw (m) and a(m) are multiplicative. If s > 1, then by the Euler product formula (See Theorem 11.7 of [3]) we have ∞ ∞ X X X Zw (m) Zw (m) Zw (m) < = a(m) s s m m ms m=1 m=1 m≤x m∈B

m∈B

Y³

´ p p + 3s + · · · 2s p p p ³ ´ Y 1 = 1 + s−1 s . p (p − 1) p =

1+

88

Xuhui Fan and Chengliang Tian

Note that if m ∈ B, then Zw (m) ≤ X Zw (m) ms m≤x

=

m∈B

=

√

m. Hence, if s >

3 , then we have 2

∞ X X Zw (m) Zw (m) − s m ms m>x m=1

m∈B ∞ X m=1

m∈B

=

No. 1

Y³

m∈B

³X 1 ´ Zw (m) + O s− 12 ms m>x m 1+

p

´ ³ 3 ´ 1 + O x 2 −s . ps−1 (ps − 1)

This proves Lemma 3.

§3. Proof of the theorem In this section, we shall use the elementary method to complete the proof of the theorem. (m − 1)m m(m + 1) Note that if +1 ≤ n ≤ , then Z(n) = m. That is, the equation 2 2 Z(n) = m has m solutions as follows: (m − 1)m m(m + 1) (m − 1)m + 1, + 2, · · · , 2 2 2 √ Since n ≤ x, from the definition of Z(n) we know that if Z(n) = m, then 1 ≤ m ≤ 8x + 1 − 1 .

2 Note that Zw (n) ≤ n, we have X X X Zw (Z(n)) = Zw (m) = m · Zw (m) + O(x) n=

n≤x

√

n≤x

m≤

Z(n)=m

=

X

√ m≤ 2x

8x+1−1 2

m · Zw (m) + O(x).

(2)

√ We separate all integer m in the interval [1, 2x] into three subsets A, B, and C as follows: A: the set of all square-free integers; B: the set of all square-full integers; C: the set of all √ S positive integer m such that m ∈ [1, 2x]/A B. Note that (2), we have X X X X Zw (Z(n)) = m · Zw (m) + m · Zw (m) + m · Zw (m) + O(x).

(3) n≤x

m≤

√

2x

m≤

m∈A

√

2x

m≤

m∈B

√

2x

m∈C

From Property 2 and Lemma 2 we know that if m ∈ A, then we have √ ³ 5´ X X 4 2 3 2 m · Zw (m) = m = 2 x2 + O x4 . π √ √ m≤

2x

m≤

m∈A

(4)

2x

m∈A

√ It is clear that if m ∈ B, then Zw (m) ≤ m. Hence X X 5 3 m · Zw (m) ¿ m2 ¿ x4 . m≤

√

2x

m∈B

m≤

√

2x

m∈B

(5)

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If m ∈ C, then we write m as m = q · n, where q is a square-free integer and n is a square-full integer. From Property 1, Property 2, Lemma 2 and Lemma 3 we have X X X m · Zw (m) = nZw (n)a(n) q 2 µ2 (q) m≤

√

√ n≤ 2x

2x

m∈C

√

q≤

2x n

à √ à 5 !!

3 4 2 x2 x4 nZw (n)a(n) · 3 +O = 5 2 π n √ n2 n≤ 2x ! Ã √ X Zw (n)a(n) 5 4 2 3 X Zw (n)a(n) = x2 + O x4 3 π2 n2 √ √ n2 n≤ 2x n≤ 2x √ ´ 3 ³ 5´ 4 2 Y³ 1 2 + O x4 = x .

1 + π2 p p(p2 − 1) X

(6)

Combining (3), (4), (5) and (6), we may immediately deduce the asymptotic formula ´ 4√ 2 ³ ³ 5´ X Y 3 1 · 2 · x2 + O x4 .

Zw (Z(n)) = 1 + (1 + 2 p(p − 1) π p n≤x

This completes the proof of Theorem.

References [1] Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, Lupton USA, American Research Press, 2000.

[2] D.S.Mitrinovic, Handbook of Number Theory, Boston London, Kluwer Academic Publishers, 1996.

[3] Tom M Apostol, Introduction to Analytic Number Theory, New York, Spinger-Verlag, 1976.

[4] Pan Chengdong and Pan Chengbiao, The elementary proof of the prime theorem, Shanghai Science and Technology Press, Shanghai, 1988.