ON A NUMBER SET RELATED TO THE K-FREE NUMBERS

ON A NUMBER SET RELATED TO THE K -FREE NUMBERS Li Congwei College of Electron and Information, Northwestern Ploytechnical University, Xi’an, Shaanxi, P.R.China

Abstract

Let Fk denotes the set of k-free number. For any positive integers l ≥ 2, we define a number set Ak,l as follows Ak,l = {n : n = ml + r, ml ≤ n < (m + 1)l , r ∈ Fk , n ∈ N }.

In this paper, we study the arithmetical properties of the number set Ak,l , and give some interesting asymptotic formulae for it.

Keywords:

Number set; k-free number; Asymptotic formula.

§1. Introduction Let k ≥ 2 be an integer. The k-free numbers set Fk is defined as follows Fk = {n : if prime p|n then pk †n, n ∈ N }.

In problem 31 of [1], Professor F.Smarandache asked us to study the arithmetical properties of the numbers in Fk . About this problem, many authors had studied it, see [2], [3], [4]. For any positive integer n and l ≥ 2, there exist an integer m such that ml ≤ n ≤ (m + 1)l .

So we can define the following number set Ak,l : Ak,l = {n : n = ml + r, ml ≤ n < (m + 1)l , r ∈ Fk , n ∈ N }.

In this paper, we use the elementary methods to study the asymptotic properties of the number of integers in Ak,l less than or equal to a fixed real number x, and give some interesting asymptotic formulae. That is, we shall prove the following results: Theorem 1. Let k, l ≥ 2 be any integers. Then for any real number x > 1, we have the asymptotic formula X n≤x n∈Ak,l

1=

³ 1 1 1´ x + Ok,l x l + k − kl , ζ(k)

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SCIENTIA MAGNA VOL.1, NO.1

where ζ(s) denotes the Riemann zeta function and Ok,l means the big Oh constant related to k, l.

Theorem 2. Assuming the Riemann Hypothesis, there holds X

1=

n≤x n∈A2,2

³ 29 ´ 6 x + O x 44 +² , π2

where ² is any fixed positive number.

§2. Two Lemmas Lemma 1. For any real number x > 1 and integer k ≥ 2, we have the asymptotic formula ³ 1´ X x + O xk .

1= ζ(k) n≤x n∈Fk

Proof. See reference [5].

Lemma 2. Assuming the Riemann Hypothesis, we have X

1=

n≤x n∈F2

³ 7 ´ 6 x + O x 22 + ² .

2 π

Proof. See reference [6].

§3. Proof of the theorems In this section, we shall complete the proofs of the theorems. For any real number x ≥ 1 and integer l ≥ 2, there exist a positive integer M such that M l ≤ x < (M + 1)l .

(1)

So from the definition of the number set Ak,l and Lemma 1, we can write X

1=

n≤x n∈Ak,l

=

M −1 X t=1

l l M −1 (t+1) X X−t

t=1

m=1 m∈Fk

(t + 1)l − tl +O ζ(k)

1

X

1

m≤x−M l m∈Fk

ÃM −1 X ³

(t + 1)l − tl

´1

!

k

t=1

µ³ ´1 ¶ x − Ml + + O x − Ml k ζ(k)

=

M −1 X t=1

=

´ ³ l−1 (t + 1)l − tl x − M l + + Ok,l M 1+ k ζ(k) ζ(k)

´ ³ l−1 x + Ok,l M 1+ k , ζ(k)

(2)

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On a number set related to the k -free numbers

On the other hand, from (1) we have the estimates 0 ≤ x − M l < (M + 1)l − M l ¿ x

l−1 l

(3)

Now combining (2) and (3), we have X

1=

n≤x n∈Ak,l

³ 1 1 1´ x + Ok,l x l + k − kl . ζ(k)

This completes the proof of Theorem 1. From the same argue as proving Theorem 1 and Lemma 2, we can get X n≤x n∈A2,2

1= =

12 π2

PM −1 t=1

= =

t+O

6 M2 π2 6 x π2

³P

7 M −1 22 +² t=1 t

³

29

+ O M 22 +² ³

+ Ok,l x

29 +² 44

´

(4)

´

(5)

´

.

(6)

This completes the proof of Theorem 2.

References [1] F.Smarandache, Only Problems, Not Solutions, Chicago, Xiquan Publ.

House, 1993, pp. 27.

[2] Zhang Tianping, On the k-Power Free Number Sequence, Smarandache Notions Journal, 14 (2004), 62-65.

[3] Zhu Weiyi, On the k-Power Complement and k-Power free Number Sequence, Smarandache Notions Journal, 14 (2004), 66-69.

[4] R.C.Baker and J.Pintz, The distribution of square free numbers, Acta Arith, 46 (1985), 71-79.