On the Pseudo-Smarandache-Squarefree function and Smarandache function

Scientia Magna Vol.4 (2008), No. 4, 7-11

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

Keywords Pseudo-Smarandache-Squarefree functionZw (n), Smarandache function S(n), me an value, asymptotic formula.

§1. Introduction and result For any positive integer n, the famous Smarandache function S(n) is defined as S(n) = min{m : n|m!, m ∈ N }, 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. For any positive integer k > 1 and prime p, we have Zw (pk ) = p.

Property 2. For any positive integer n, we have Zw (n) ≤ n.

Property 3. The function Zw (n) is multiplicative. That is, if GCD(m, n) = 1, then Zw (m· n) = Zw (m) · Zw (n).

The main purpose of this paper is using the elementary methods to study the mean value properties of Zw (S(n)) and S(n) · Zw (n), and give two sharper asymptotic formulas for it. That is, we shall prove the following conclusions: Theorem 1. Let k ≥ 2 be any fixed positive integer. Then for any real number x ≥ 2,

8

Xuhui Fan

No. 4

we have the asymptotic formula X

³ x2 ´ X ci · x2 π 2 x2 · + + O , 12 ln x i=2 lni x lnk+1 x k

Zw (S(n)) =

n≤x

where ci (i = 2, 3 · · · k) are computable constants.

Theorem 2. Let k ≥ 2 be any fixed positive integer. Then for any real number x ≥ 2, we have the asymptotic formula X

³ x3 ´ X ei · x3 ζ(2) · ζ(3) Y ³ 1 ´ x3 1− · + + O , 3ζ(4) p + p3 ln x i=2 lni x lnk+1 x p k

Zw (n) · S(n) =

n≤x

where ζ(n) is the Riemann zeta-function,

Y

denotes the product over all primes, ei (i =

p

2, 3 · · · k) are computable constants.

§2. A simple lemma To complete the proof of the theorem, we need the following : Lemma. For any real number x ≥ 2 and s ≥ 2 , we have the asymptotic formula ´ ³ X Zw (n) ζ(s) · ζ(s − 1) Y ³ 1 ´ 1− 2s = · .

1 − + O x ns ζ(2(s − 1)) p + ps √ p n≤ x

Proof. Note that Property 1 and 3, by the Euler product formula (See Theorem 11.7 of [2]), we have ∞ X Zw (n) ns n=1

´ Zw (p) Zw (p2 ) + + ··· s 2s p p p ´ Y³ p p = 1 + s + 2s + · · · p p p ´ Y³ 1 1 = 1 + s−1 · −s p 1−p p 1 ´ ζ(s)ζ(s − 1) Y ³ 1− = . ζ(2(s − 1)) p p + ps =

Y³

1+

From (1) we have ∞ ³ ´ X Zw (n) X Zw (n) X Zw (n) ζ(s)ζ(s − 1) Y ³ 1 ´ 1− 2s = − = + O x 1 − . ns ns ns ζ(2(s − 1)) p p + ps √ √ n=1 n> x

n≤ x

Specially, if s = 3, then we have the asymptotic formula ³ 1 ´ X Zw (n) ζ(3)ζ(2) Y ³ 1 ´ √ . = + O 1 − ns ζ(4) p + p3 x √ p n≤ x

This completes the proof of Lemma.

(1)

Vol. 4

On the Pseudo-Smarandache-Squarefree function and Smarandache function

9

§3. Proof of the theorems In this section, we shall use the elementary methods to complete the proof of the theorems.

First we prove Theorem 1. We separate all integer n in the interval [1, x] into two subsets A and B as follows: √ A: p | n and p > n, where p is a prime. B: other positive integer n such that n ∈ [1, x]\A. From the definition of the subsets A and B we have X

Zw (S(n)) =

X

Zw (S(n)) +

n∈A

n≤x

X

Zw (S(n)).

(2)

n∈B

From Property 1 and the definition of the function S(n) and the subset A we know that if n ∈ A, then we have X

Zw (S(n)) =

n∈A

X

Zw (S(pn)) =

pn≤x

X

Zw (p) =

pn≤x

p>n

X

p=

pn≤x

p>n

p>n

X

X

√ x n≤ x n<p≤ n

p.

(3)

By the Abel’s summation formula (see Theorem 4.2 of [2]) and the Prime Theorem (see Theorem 3.2 of [3]): π(x) =

X

1=

p≤x

k X ai · x i=1

lni x

³ +O

x lnk+1 x

´ ,

where ai (i = 1, 2, 3 · · · k) are computable constants and a1 = 1, we have X

p

=

x n<p≤ n

x ³x´ π − nπ(n) − n n

Z

x n

π(t)dt

n

´ ³ X bi · x2 · lni n x2 x2 , + +O i k+1 2 2n · ln x i=2 n2 · ln x n2 · ln x k

=

(4)

where bi (i = 2, 3 · · · k) are computable constants. ∞ ∞ X X 1 lni n π2 Note that = and is convergent for all i = 1, 2, · · · , k. Combining (3) n2 6 n2 n=1 n=1 and (4) we have X

Zw (S(n)) =

n∈A

X √ n≤ x

Ã

à !! k X x2 bi · x2 · lni n x2 +O + 2n2 · ln x i=2 n2 · lni x n2 lnk+1 x

³ x2 ´ X ci · x2 π 2 x2 + O · + , 12 ln x i=2 lni x lnk+1 x k

=

(5)

where ci (i = 2, 3 · · · k) are computable constants. αs 1 α2 Now we estimate the error terms in set B. Let n = pα 1 p2 · · · ps be the factorization of n into prime powers. If n ∈ B, then we have ´ ³ √ 5 i S(n) = max S(pα n ln n ¿ n 6 .

(6) i ) ≤ max (αi pi ) ≤ 1≤i≤s

1≤i≤s

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Xuhui Fan

No. 4

From (6) and Property 2 we have X

Zw (S(n)) ¿

n∈B

X

5

11

n6 ¿ x 6 .

(7)

n≤x

Combining (2), (5) and (7) we have X

³ x2 ´ X ci · x2 π 2 x2 · + + O .

12 ln x i=2 lni x lnk+1 x k

Zw (S(n)) =

n≤x

This proves Theorem 1.

Now we prove Theorem 2. From Property 1 and Property 3 we have X

S(n) · Zw (n) =

n∈A

X

S(pn) · Zw (pn) =

pn≤x

n<p

X

p2

=

x n<p≤ n

x2 ³ x ´ π − n2 π(n) − 2 n2 n

X

X

√ x n≤ x n<p≤ n

Z

x n

p2 · Zw (n).

(8)

t · π(t)dt

n

³ ´ X di · x3 · lni n x3 x3 + +O , i k+1 3 3n · ln x i=2 n3 · ln x n3 · ln x k

=

(9)

where di (i = 2, 3 · · · k) are computable constants. ∞ X lni n · Zw (n) is convergent for all i = 1, 2, · · · , k. Combining Note that the lemma and n3 n=1 (8) and (9) we have X

S(n) · Zw (n)

=

n∈A

X √ n≤ x

Ã

à !! k X di · x3 · lni n x3 x3 + +O · Zw (n) 3n3 · ln x i=2 n3 · lni x n3 · lnk+1 x

³ x3 ´ X ei · x3 1 ´ x3 ζ(2)ζ(3) Y ³ 1− · + , + O 3ζ(4) p p + p3 ln x i=2 lni x lnk+1 x k

=

(10)

where ei (i = 2, 3 · · · k) are computable constants.

If n ∈ B, then we have X

S(n) · Zw (n) ¿

n∈B

X√

5

n ln n · n ¿ x 2 ln x.

n≤x

Combining (10) and (11) we have X n≤x

³ x3 ´ X ei · x3 1 ´ x3 ζ(2) · ζ(3) Y ³ · + 1− + O .

Zw (n) · S(n) = i 3ζ(4) p + p3 ln x i=2 ln x lnk+1 x p

This proves Theorem 2.

k

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On the Pseudo-Smarandache-Squarefree function and Smarandache function

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

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

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