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.