On the mean value of the F.Smarandache simple divisor function

Scientia Magna Vol. 2 (2006), No. 1, 35-37

On the mean value of the F.Smarandache simple divisor function Yang Qianli Department of Mathematics, Weinan Teacher’s College Weinan, Shaanxi, P.R.China Abstract In this paper, we introduce a new arithmetic function τsp (n) which we called the simple divisor function. The main purpose of this paper is to study the asymptotic properties of the mean value of τsp (n) by using the elementary methods, and obtain an interesting asymptotic formula for it.

Keywords mula.

Smarandache simple number divisor; Simple divisor function; Asymptotic for-

§1. Introduction A positive integer n is called simple number if the product of its all proper divisors is less than or equal to n. In problem 23 of [1], Professor F.Smarandache asked us to study the properties of the sequence of the simple numbers. About this problem, many scholars have studied it before. For example, in [2], Liu Hongyan and Zhang Wenpeng studied the mean value properties of 1/n and 1/φ(n) (where n is a simple number), and obtained two asymptotic formulae for them. For convenient, let A denotes the set of all simple numbers, they proved that µ ¶ X 1 ln ln x = (ln ln x)2 + B1 ln ln x + B2 + O n ln x n∈A n≤x

and

X n∈A n≤x

1 = (ln ln x)2 + C1 ln ln x + C2 + O φ(n)

µ

ln ln x ln x

¶ ,

where B1 , B2 , C1 , C2 are constants, and φ(n) is the Euler function. αk 1 α2 For n > 1, let n = pα 1 p2 · · · pk denotes the factorization of n into prime powers. If one of the divisor d of n satisfing τ (d) ≤ 4 (where τ (n) denotes the numbers of all divisors of n), then we call d as a simple number divisor. In this paper, we introduce a new arithmetic function X τsp (n) = 1, d|n τ (d)≤4

which we called the simple divisor function. The main purpose of this paper is to study the asymptotic property of the mean value of τsp (n) by using the elementary methods, and obtain an interesting asymptotic formula for it. That is, we will prove the following:

36

Yang Qianli

No. 1

Theorem. For any real number x ≥ 1, we have the asymptotic formula µ ¶ µ ¶ X 1 1 b+A x log log x 2 τsp (n) = x(log log x) + (a + 1)x log log x + +B+C x+O , 2 2 2 log x

n≤x

where a and b are two computable constants, A = γ + constant, B =

X

(log(1 − 1/p) + 1/p), γ is the Euler

p

X 1 X 1 and C = . p2 p3 p p

§2. Two Lemmas Before the proof of Theorem, two useful Lemmas will be introduced which we will use subsequently.

Lemma 1. For any real number x ≥ 1, we have the asymptotic formula µ ¶ X x (a) ω(n) = x log log x + Ax + O , log x n≤x

X

(b) where A = γ +

X

µ 2

2

ω (n) = x(log log x) + ax log log x + bx + O

n≤x

x log log x log x

¶ ,

(log(1 − 1/p) + 1/p), γ is the Euler constant, a and b are two computable

p

constants.

Proof. See references [3] and [4].

Lemma 2. For any positive integer n ≥ 1, we have X X 1 1 τsp (n) = ω 2 (n) + ω(n) + 1+ 1, 2 2 2 3 p |n

p |n

where ω(n) denotes the number of all different prime divisors of n, of all primes such that p2 | n,

X

X

1 denotes the number

p2 |n

1 denotes the number of all primes such that p3 | n.

p3 |n

αk 1 α2 Proof. Let n > 1, we can write n = pα 1 p2 · · · pk , then from the definition of τsp (n) we know that there are only four kinds of divisors d such that the number of the divisors of d less or equal to 4. That is p | n, pi pj | n, p2 | n and p3 | n, where pi 6= pj .

Hence, we have X X X X τsp (n) = 1+ 1+ 1+ 1 p|n

pi pj |n pi 6=pj

p2 |n

p3 |n

X X 1 = ω(n) + ω(n)(ω(n) − 1) + 1+ 1 2 2 3 p |n

=

This proves Lemma 2.

1 2 1 ω (n) + ω(n) + 2 2

X p2 |n

1+

X p3 |n

p |n

1

Vol. 2

On the mean value of the F.Smarandache simple divisor function

37

§3. Proof of the theorem Now we completes the proof of Theorem. From the definition of the simple divisor function, Lemma 1 and Lemma 2, we can write   X X X X 1  ω 2 (n) + 1 ω(n) + τsp (n) = 1+ 1 2 2 2 3 n≤x

n≤x

p |n

p |n

XX XX 1X 2 1X = ω (n) + ω(n) + 1+ 1 2 2 n≤x n≤x n≤x p2 |n n≤x p3 |n µ µ ¶¶ x log log x 1 = x(log log x)2 + ax log log x + bx + O 2 log x µ µ ¶¶ X · ¸ X · ¸ 1 x x x + x log log x + Ax + O + + 2 2 log x p p3 p≤x p≤x µ µ ¶¶ x log log x 1 = x(log log x)2 + (a + 1)x log log x + (b + A)x + O 2 log x µ ¶ µ ¶ X 1 X 1 x x +O +x +O + x p2 log x p3 log x p≤x p≤x µ µ ¶¶ 1 x log log x = x(log log x)2 + (a + 1)x log log x + (b + A)x + O 2 log x µ ¶ x + (B + C)x + O log x µ ¶ µ ¶ 1 b+A x log log x 1 x(log log x)2 + (a + 1)x log log x + +B+C x+O . = 2 2 2 log x X 1 X 1 and C = . p2 p3 p p This completes the proof of Theorem.

where B =

References [1] F. Smarandache, Only problems, not Solutions, Xiquan Publ. House, Chicago, 1993, pp. 23.

[2] Liu Hongyan and Zhang Wenpeng, On the simple numbers and the mean value properties, Smarandache Notions Journal, 14(2004), pp. 171.

[3] G.H.Hardy and S.Ramanujan, The normal number of prime factors of a number n, Quart. J. Math., 48(1917), pp. 76-92.

[4] H.N.Shapiro, Introduction to the theory of numbers, John Wiley and Sons, 1983, pp.

347.