The mean value of a new arithmetical function

Scientia Magna Vol. 4 (2008), No. 1, 79-82

The mean value of a new arithmetical function Jin Zhang†‡ and Pei Zhang † ‡

†

Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China Department of Mathematics, Xi’an Normal School, Xi’an, Shaanxi, P.R.China

Abstract The main purpose of this paper is using the elementary and the analytic methods to study the mean value properties of a Smarandache multiplicative function, and give two sharper asymptotic formulae for it.

Keywords

Smarandache multiplicative function, mean value, asymptotic formula.

§1. Introduction For any positive integer n, we call an arithmetical function f (n) as the Smarandache multiplicative function if for any positive integers m and n with (m, n) = 1, we have f (mn) = max{f (m), f (n)}. For example, the Smarandache function S(n) and the Smarandache LCM function SL(n) both are Smarandache multiplicative functions. Now we define a new ¾ ½ Smaran1 dache multiplicative function f (n) as follows: f (1) = 1; If n > 1, then f (n) = max , 1≤i≤k αi + 1 αk α1 α2 where n = p1 p2 · · · pk be the factorization of n into prime powers. The first few values of 1 1 1 1 1 1 1 f (n) are f (1) = 1, f (2) = , f (3) = , f (4) = , f (5) = , f (6) = , f (7) = , f (8) = , 2 2 3 2 2 2 4 1 1 1 f (9) = , f (10) = , f (11) = , · · · · · · .

3 2 2 1 . About the Generally, for any prime p and positive integer α, we have f (pα ) = 1+α elementary properties of f (n), it seems that none had studied it before. This function is interesting, because its value only depend on the power of primes. The main purpose of this paper is using the elementary and the analytic methods to study the mean value properties of f (n), and give two sharper asymptotic formulas for it. That is, we shall prove the following: Theorem 1. For any real number x > 1, we have the asymptotic formula X n≤x

f (n) =

³ x ´ 1 , x ln ln x + c · x + O 2 ln x

where c is a computable constant.

Theorem 2. For any real number x > 1, we also have the asymptotic formula ¡ ¢ ¶2 ³ 1´ Xµ √ 1 ζ 32 √ 1 = f (n) − · x · ln ln x + d · x + O x 3 , 2 36 ζ(3) n≤x

where ζ(s) is the Riemann zeta-function, and d is a computable constant.

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Jin Zhang and Pei Zhang

No. 1

§2. Proof of the theorems In this section, we shall using the elementary and the analytic methods to prove our Theorems. First we give following two simple lemmas: Lemma 1. Let A denotes the set of all square-full numbers. Then we have the asymptotic formula ¡ ¢ ¡ ¢ ³ 1´ X ζ 23 ζ 32 1 1 2 ·x + · x3 + O x6 , 1= ζ(3) ζ(2) n≤x n∈A

where ζ(s) is the Riemann zeta-function.

Lemma 2. Let B denotes the set of all cubic-full numbers. Then we have ³ 1´ X 1 1 = N · x3 + O x4 , n≤x n∈B

where N is a computable constant.

Proof. The proof of these two Lemmas can be found in reference [3].

Now we use these two simple Lemmas to complete the proof of our Theorems. In fact, for αk 1 α2 any positive integer n > 1, we can write it as n = pα 1 p2 · · · pk , then from the definition of f (n), we have X X X f (n) = f (n) + f (n), n≤x

n≤x n∈A

n≤x n∈B

where A denotes the set of all square-full numbers. That is, n > 1, and for any prime p, if p | n, then p2 | n. B denotes the set of all positive integers with n ∈ / A. Note that f (n) ¿ 1, from the definition of A and Lemma 1 we have ³ 1´ X (1) f (n) = O x 2 . n≤x n∈A

X

f (n) =

n≤x n∈B

X np≤x (n, p)=1

=

f (n) =

X p≤x

X n≤ x p

1 2

(n, p)=1

µ ¶ 1X x x − 2 + O(1) 2 p p p≤x

=

  X xX1 xX 1 1 +O − 1 .

2 p 2 p2 2 p≤x

p≤x

p≤x

Note that µ ¶ X1 1 = ln ln x + c + O ( see Theorem 4.12 of reference [2] ), p ln x p≤x

µ ¶ X 1 X 1 X 1 1 = − = d + O , 2 2 2 p p p x p p>x

p≤x

(2)

Vol. 4

The mean value of a new arithmetical function

81

where c and d are two computable constants.

And the Prime Theorem ( see Theorem 3.2 of reference [3]):

π(x) =

X p≤x

x +O 1= ln x

µ

x ln2 x

¶ .

So from (2) we have X

f (n)

=

n≤x n∈B

= =

x 2

µ

µ ln ln x + c + O

1 ln x

¶¶ −

x 2

µ d+O

µ ¶¶ µ µ µ ¶¶¶ 1 1 x x +O +O x 2 ln x ln2 x

³ x ´ 1 c d x ln ln x + x − x + O 2 2 2 ln x ³ x ´ 1 x ln ln x + λx + O , 2 ln x

(3)

where λ is a computable constant.

Now combining (1) and (3) we may immediately get X

f (n) =

1+

n≤x

X

f (n) +

n≤x n∈A

X

f (n)

n≤x n∈B

³ 1´ 1 ³ x ´ = 1 + O x 2 + x ln ln x + λ · x + O 2 ln x ³ x ´ 1 = x ln ln x + λ · x + O , 2 ln x where λ is a computable constant.

This proves Theorem 1.

Now we complete the proof of Theorem 2. From the definition of f (n) and the properties of square-full numbers, we have Xµ n≤x

f (n) −

1 2

¶2 =

µ ¶2 X µ ¶2 1 X 1 1 + f (n) − + f (n) − 4 2 2 n≤x n∈A

=

µ ¶2 1 1 X + f (n) − .

4 2

n≤x n∈A /

n≤x n∈A

where A also denotes the set of all square-full numbers. Let C denotes the set of all cubic-full

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Jin Zhang and Pei Zhang

No. 1

numbers. Then from the properties of square-full numbers, Lemma 1 and Lemma 2 we have Xµ n≤x n∈A

1 f (n) − 2

¶2

µ

X

=

np2 ≤x (n, p)=1, n∈A

=

X

X

p2 ≤x

n≤ px2

1 f (n) − 2 µ

1 1 − 3 2

(n, p)=1, n∈A

 =

n≤ p2

=

n≤x n∈C

¶2

1 f (n) − 2



¶2



X  +O 1   n≤x n∈C



n≤ p4

n∈A

=

+

Xµ

³ 1´ X X 1 X 1  −   + O x3  x 36 x 36  2

p ≤x

=

¶2

Ã

n∈A

!

1 ³ 1´ x x2 c· − c · 2 + O x3 p p p ≤x µ ¶ ³ 1´ X 1 c √ 1 · x· − 2 + O x3 36 p p √ p≤ x ³ 1´ √ √ 1 c · x · ln ln x + d · x + O x 3 .

36 1 X 36 2

1 2

¡ ¢ ζ 32 where c = , d is a computable constant. ζ(3) So we have the asymptotic formula ¡ ¢ ¶2 ³ 1´ Xµ √ 1 1 ζ 32 √ f (n) − · x · ln ln x + d · x + O x 3 . = 2 36 ζ(3) n≤x

This completes the proof of Theorem 2.

References [1] F.Smarandache, Only problems, Not solutions, Xiquan Publishing House, Chicago, 1993.

[2] Tom M. Apostol, Introduction to Analytical Number Theory, Spring-Verlag, New York, 1976.

[3] Aleksandar Ivi´c, The Riemann Zeta-Function, Dover Publications, New York, 2003.

[4] Zhefeng Xu, On the k-full Number Sequences, Smarandache Notions Journal, 14(2004), 159-163.

[5] Jinping Ma, The Smarandache multiplicative function, Scientia Magna, 1(2005), No.1, 125-128.