On the Smarandache function and the divisor product sequences

Scientia Magna Vol. 5 (2009), No. 2, 129-132

On the Smarandache function and the divisor product sequences Mingdong Xiao Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China

Abstract Let n be any positive integer, Pd (n) denotes the product of all positive divisors of n. The main purpose of this paper is using the elementary and analytic methods to study the mean value properties of a new arithmetical function S (Pd (n)), and give an interesting asymptotic formula for it.

Keywords Smarandach function, Divisor product sequences, Composite function, mean value, Asymptotic formula.

§1. Introduction For any positive integer n, the famous F.Smarandache function S(n) is defined as the smallest positive integer m such that n divide m!. That is, S(n) = min{m : m ∈ N, n|m!}.

And the Smarandache divisor product sequences {Pd (n)} is defined as the product of all positive d(n) Y divisors of n. That is, Pd (n) = d = n 2 , where d(n) is the Dirichlet divisor function. d|n

For examples, Pd (1) = 1, Pd (2) = 2, Pd (3) = 3, Pd (4) = 8, · · · . In problem 25 of reference [1], Professor F.Smarandache asked us to study the properties of the function S(n) and the sequence {Pd (n)}. About these problems, many scholars had studied them, and obtained a series interesting results, see references [2], [3], [4], [5] and [6]. But at present, none had studied the mean value properties of the composite function S (Pd (n)), at least we have not seen any related papers before. In this paper, we shall use the elementary methods to study the mean value properties of S(Pd (n)), and give an interesting asymptotic formula for it. That is, we shall prove the following conclusion: Theorem. For any fixed positive integer k and any real number x ≥ 1, we have the asymptotic formula X

k

X π 4 x2 x2 S(Pd (n)) = · + bi · i + O 72 ln x i=2 ln x n≤x where bi (i = 2, 3, · · · , k) are computable constants.

µ

x2 lnk+1 x

¶ ,

130

Mingdong Xiao

No. 2

§2. Some simple lemmas To complete the proof of the theorem, we need the following several simple lemmas. First we have Lemma 1. For any positive integer α, we have the estimate S(pα ) ≤ αp.

Especially, when α ≤ p, we have S(pα ) = αp, where p is a prime.

Proof. See reference [3]. αk 1 α2 Lemma 2. For any positive integer n, let n = pα 1 p2 · · · pk denotes the factorization of n into prime powers, then we have i S(n) = max {S(pα i )}.

1≤i≤k

√ Lemma 3. Let P (n) denotes the greatest prime divisor of n, if P (n) > n, then we have S(n) = P (n).

Proof. The proof of Lemma 2 and Lemma 3 can be found in reference [4].

§3. Proof of the theorem In this section , we shall use the above lemmas to complete the proof of our theorem. For any positive integer n, it is clear that from the definition of Pd (n) we have X     1 Y Yn 2 r|n =n Pd (n) =  r ·  = nd(n) . r r|n

r|n

d(n)

αk 1 α2 So we have the identity Pd (n) = n 2 . Let n = pα 1 p2 · · · pk denotes the factorization of n into prime powers. First we separate all integers n in the interval [1, x] into two subsets A and B as follows:

A = {n : n ≤ x, P (n) ≤

√

n}, B = {n : n ≤ x, P (n) >

√

If n ∈ A, then from Lemma 1 and Lemma 2, and note that Pd (n) = n

n}.

d(n) 2

we have

αk d(n) α1 d(n) α2 d(n) d(n) · · · pk 2 .

Pd (n) = n 2 = p1 2 p2 2 Therefore,     αk d(n) αi d(n)  α1 d(n) α2 d(n)        · · · pk 2  = max S pi 2  S (Pd (n)) = S p1 2 p2 2  1≤i≤k    ½ ¾ αi d(n) d(n) √ ≤ max pi ≤ n ln n.

1≤i≤k 2 2 

Vol. 5

On the Smarandache function and the divisor product sequences

131

From reference [10] we know that X

d(n) = x ln x + O(x).

n≤x

So we have the estimate X X d(n) √ X √ 3 S (Pd (n)) ≤ n ln n ¿ d(n) x ln x ¿ x 2 ln2 x.

2 n∈A

n∈A

(1)

n≤x

√ √ If n ∈ B, let n = n1 p, where n1 < n < p. It is clear that d(n1 ) < n < p and d(n) = 2d(n1 ). So from Lemma 3 we have     d(n1 p) d(n1 p) X X X S (Pd (n)) = S (n1 p) 2  = S p 2  n∈B

n1 p≤x n1 <p

=

X

X

√ x n≤ x n<p≤ n

X

d(n)p =

√ n≤ x

n1 p≤x n1 <p

d(n)

X

n<p≤

 =

=

X √ n≤ x

X √ n≤ x

X

d(n)

x n X

p+O

√ n≤ x

p≤

d(n)

p≤

X

d(n) ·

p x n



n  ln n

p + O(x).

(2)

x n

From the Abel’s summation formula (see Theorem 4.2 of [10]) and the Prime Theorem (see Theorem 3.2 of [11]) we have π(x) =

k X ai · x i=1

lni x

µ

x

+O

lnk+1 x

¶ ,

where ai (i = 1, 2, · · · , k) are computable constants and a1 = 1. We have Z nx X x ³x´ p = π − π(y)dy n n 2 x p≤ n µ ¶ k X x2 x2 ci · x2 lni n = +O , + 2n2 ln x i=2 n2 ln2 x n2 lnk+1 x

(3)

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

Note that ∞ X 1 π2 = n2 6 n=1 and ∞ X d(n) = n2 n=1

Ã

∞ X 1 2 n n=1

!2 =

π4 , 36

(4)

132

Mingdong Xiao

No. 2

from (2), (3) and (4) we obtain X

S (Pd (n)) =

n∈B

µ ¶ k X X x2 X d(n) ci · x2 d(n) lni n x2 + + O 2 ln x √ n2 √ n2 lni x lnk+1 x i=2 n≤ x

4

=

2

n≤ x

k X

x2 π x + bi · i + O 72 ln x i=2 ln x

µ

x2

¶

lnk+1 x

,

(5)

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

Now combining (1) and (5) we may immediately get the asymptotic formula X X X S (Pd (n)) = S (Pd (n)) + S (Pd (n)) n∈A

n≤x

=

n∈B k X

π 4 x2 x2 · + bi · i + O 72 ln x i=2 ln x

µ

x2 lnk+1 x

¶ ,

where bi (i = 2, 3, · · · , k) are computable constants. This completes the proof of Theorem.

References [1] F. Smarandache, Only Problems, Not Solutions, Xiquan Publishing House, Chicago, 1993.

[2] Liu Hongyan and Zhang Wenpeng, On the divisor products and proper divisor products sequences, Smarandache Notions Journal, 13(2002), 128-133.

[3] F.Mark, M.Patrick, Bounding the Smarandache function, Smarandache Notions Journal, 13(2002), 37-42.

[4] Wang Yongxing, On the smarandache function, Research on Smarandache problems in number theory , Hexis, (2005), 103-106.

[5] Wu Qibin, A composite function involving the Smarandache function, Pure and Applied Mathematics, 23(2007), No.4, 463-466.

[6] Liu Yanni, Li Ling and Liu Baoli, Smarandache Unsolved Problems and New Progress, High American Press, 2008.

[7] Yi Yuan and Kang Xiaoyu, Research on Smarandache Problems, High American Press, 2006.

[8] Zhang Wenpeng, The elementary number theory (in Chinese), Shaanxi Normal University Press, Xi’an, 2007.

[9] Chen guohui, New Progress On Smarandache Problems, High American Press, 2007.

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

[11] Pan Chengdong and Pan Chengbiao, The elementary proof of the Prime Theorem, Shanghai Science and Technology Press, Shanghai, 1988.