On the Pseudo-Smarandache-Squarefree function and Smarandache function

2008·0.21 MB·

by  Xuhui Fan

Scientia Magna
Vol.4 (2008), No. 4, 7-11
On the Pseudo-Smarandache-Squarefree
function and Smarandache function
Xuhui Fan
†Foundation Department, Engineering College of Armed Police Force,
Xi’an, Shaanxi, 710086
‡ Department of Mathematics, Northwest University, Xi’an, Shaanxi, 710069
Abstract For any positive integer n, Pseudo-Smarandache- Squarefree function Z (n) is
w
defined as Z (n) = min{m : n|mn,m ∈ N}. Smarandache function S(n) is defined as
w
S(n) = min{m : n|m!,m ∈ N}. The main purpose of this paper is using the elementary
methods to study the mean value properties of the Pseudo-Smarandache-Squarefree function
and Smarandache function, and give two sharper asymptotic formulas for it.
Keywords Pseudo-Smarandache-Squarefree functionZ (n), Smarandache function S(n), me
w
an value, asymptotic formula.
§1. Introduction and result
For any positive integer n, the famous Smarandache function S(n) is defined as S(n) =
min{m : n|m!,m ∈ N}, Pseudo-Smarandache-Squarefree function Z (n) is defined as the
w
smallest positive integer m such that n | mn. That is,
Z (n)=min{m:n|mn,m∈N}.
w
For example Z (1) = 1, Z (2) = 2, Z (3) = 3, Z (4) = 2, Z (5) = 5, Z (6) = 6, Z (7) = 7,
w w w w w w w
Z (8) = 2, Z (9) = 3, Z (10) = 10, ···. About the elementary properties of Z (n), some
w w w w
authors had studied it, and obtained some interesting results. For example, Felice Russo [1]
obtained some elementary properties of Z (n) as follows:
w
Property 1. For any positive integer k >1 and prime p, we have Z (pk)=p.
w
Property 2. For any positive integer n, we have Z (n)≤n.
w
Property3. ThefunctionZ (n)ismultiplicative. Thatis,ifGCD(m,n)=1,thenZ (m·
w w
n)=Z (m)·Z (n).
w w
The main purpose of this paper is using the elementary methods to study the mean value
propertiesofZ (S(n))andS(n)·Z (n),andgivetwosharperasymptoticformulasforit. That
w w
is, we shall prove the following conclusions:
Theorem 1. Let k ≥ 2 be any fixed positive integer. Then for any real number x ≥ 2,
8 XuhuiFan No. 4
we have the asymptotic formula
(cid:88) π2 x2 (cid:88)k c ·x2 (cid:179) x2 (cid:180)
Z (S(n))= · + i +O ,
w 12 lnx lnix lnk+1x
n≤x i=2
where c (i=2,3···k) are computable constants.
i
Theorem 2. Let k ≥ 2 be any fixed positive integer. Then for any real number x ≥ 2,
we have the asymptotic formula
(cid:88) ζ(2)·ζ(3)(cid:89)(cid:179) 1 (cid:180) x3 (cid:88)k e ·x3 (cid:179) x3 (cid:180)
Z (n)·S(n)= 1− · + i +O ,
w 3ζ(4) p+p3 lnx lnix lnk+1x
n≤x p i=2
(cid:89)
where ζ(n) is the Riemann zeta-function, denotes the product over all primes, e (i =
i
p
2,3···k) are computable constants.
§2. A simple lemma
To complete the proof of the theorem, we need the following :
Lemma. For any real number x≥2 and s≥2 , we have the asymptotic formula
(cid:88) Z (n) ζ(s)·ζ(s−1) (cid:89)(cid:179) 1 (cid:180) (cid:179) (cid:180)
w = · 1− +O x1−2s .
ns ζ(2(s−1)) p+ps
√
n≤ x p
Proof. Note that Property 1 and 3, by the Euler product formula (See Theorem 11.7 of
[2]), we have
(cid:88)∞ Z (n) (cid:89)(cid:179) Z (p) Z (p2) (cid:180)
w = 1+ w + w +···
ns ps p2s
n=1 p
(cid:89)(cid:179) p p (cid:180)
= 1+ + +···
ps p2s
p
(cid:89)(cid:179) 1 1 (cid:180)
= 1+ ·
ps−1 1−p−s
p
ζ(s)ζ(s−1)(cid:89)(cid:179) 1 (cid:180)
= 1− . (1)
ζ(2(s−1)) p+ps
p
From (1) we have
(cid:88) Z (n) (cid:88)∞ Z (n) (cid:88) Z (n) ζ(s)ζ(s−1)(cid:89)(cid:179) 1 (cid:180) (cid:179) (cid:180)
w = w − w = 1− +O x1−2s .
ns ns ns ζ(2(s−1)) p+ps
√ √
n≤ x n=1 n> x p
Specially, if s=3, then we have the asymptotic formula
(cid:88) Z (n) ζ(3)ζ(2)(cid:89)(cid:179) 1 (cid:180) (cid:179) 1 (cid:180)
w = 1− +O √ .
ns ζ(4) p+p3 x
√
n≤ x p
This completes the proof of Lemma.
Vol. 4 OnthePseudo-Smarandache-SquarefreefunctionandSmarandachefunction 9
§3. Proof of the theorems
Inthissection,weshallusetheelementarymethodstocompletetheproofofthetheorems.
First we prove Theorem 1. We separate all integer n in the interval [1,x] into two subsets
A and B as follows:
√
A:p|nandp> n,wherepisaprime. B:otherpositiveintegernsuchthatn∈[1,x]\A.
From the definition of the subsets A and B we have
(cid:88) (cid:88) (cid:88)
Z (S(n))= Z (S(n))+ Z (S(n)). (2)
w w w
n≤x n∈A n∈B
From Property 1 and the definition of the function S(n) and the subset A we know that if
n∈A, then we have
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
Z (S(n))= Z (S(pn))= Z (p)= p= p. (3)
w w w
√
n∈A ppn>≤nx ppn>≤nx ppn>≤nx n≤ xn<p≤nx
By the Abel’s summation formula (see Theorem 4.2 of [2]) and the Prime Theorem (see
Theorem 3.2 of [3]):
(cid:88) (cid:88)k a ·x (cid:179) x (cid:180)
π(x)= 1= i +O ,
lnix lnk+1x
p≤x i=1
where a (i=1,2,3···k) are computable constants and a =1, we have
i 1
(cid:88) x (cid:179)x(cid:180) (cid:90) nx
p = π −nπ(n)− π(t)dt
n n
n<p≤x n
n
x2 (cid:88)k b ·x2·lnin (cid:179) x2 (cid:180)
= + i +O , (4)
2n2·lnx n2·lnix n2·lnk+1x
i=2
where b (i=2,3···k) are computable constants.
i
(cid:88)∞ 1 π2 (cid:88)∞ lnin
Note that = and is convergent for all i = 1,2,··· ,k. Combining (3)
n2 6 n2
n=1 n=1
and (4) we have
(cid:195) (cid:195) (cid:33)(cid:33)
(cid:88) (cid:88) x2 (cid:88)k b ·x2·lnin x2
Z (S(n)) = + i +O
w √ 2n2·lnx n2·lnix n2lnk+1x
n∈A n≤ x i=2
π2 x2 (cid:88)k c ·x2 (cid:179) x2 (cid:180)
= · + i +O , (5)
12 lnx lnix lnk+1x
i=2
where c (i=2,3···k) are computable constants.
i
Now we estimate the error terms in set B. Let n=pα1pα2···pαs be the factorization of n
1 2 s
into prime powers. If n∈B, then we have
(cid:179) (cid:180)
√
S(n)= max S(pαii) ≤ max(αipi)≤ nlnn(cid:191)n65. (6)
1≤i≤s 1≤i≤s
10 XuhuiFan No. 4
From (6) and Property 2 we have
(cid:88) (cid:88)
Zw(S(n))(cid:191) n56 (cid:191)x161. (7)
n∈B n≤x
Combining (2), (5) and (7) we have
(cid:88) π2 x2 (cid:88)k c ·x2 (cid:179) x2 (cid:180)
Z (S(n))= · + i +O .
w 12 lnx lnix lnk+1x
n≤x i=2
This proves Theorem 1.
Now we prove Theorem 2. From Property 1 and Property 3 we have
(cid:88) (cid:88) (cid:88) (cid:88)
S(n)·Z (n)= S(pn)·Z (pn)= p2·Z (n). (8)
w w w
√
n∈A pnn<≤px n≤ xn<p≤nx
(cid:88) p2 = x2π(cid:179)x(cid:180)−n2π(n)−2(cid:90) nx t·π(t)dt
n2 n
n<p≤x n
n
x3 (cid:88)k d ·x3·lnin (cid:179) x3 (cid:180)
= + i +O , (9)
3n3·lnx n3·lnix n3·lnk+1x
i=2
where d (i=2,3···k) are computable constants.
i
(cid:88)∞ lnin·Z (n)
Note that the lemma and w is convergent for all i=1,2,··· ,k. Combining
n3
n=1
(8) and (9) we have
(cid:195) (cid:195) (cid:33)(cid:33)
(cid:88) (cid:88) x3 (cid:88)k d ·x3·lnin x3
S(n)·Z (n) = + i +O ·Z (n)
w √ 3n3·lnx n3·lnix n3·lnk+1x w
n∈A n≤ x i=2
ζ(2)ζ(3)(cid:89)(cid:179) 1 (cid:180) x3 (cid:88)k e ·x3 (cid:179) x3 (cid:180)
= 1− · + i +O , (10)
3ζ(4) p+p3 lnx lnix lnk+1x
p i=2
where e (i=2,3···k) are computable constants.
i
If n∈B, then we have
(cid:88) (cid:88)√
S(n)·Zw(n)(cid:191) nlnn·n(cid:191)x52 lnx. (11)
n∈B n≤x
Combining (10) and (11) we have
(cid:88) ζ(2)·ζ(3)(cid:89)(cid:179) 1 (cid:180) x3 (cid:88)k e ·x3 (cid:179) x3 (cid:180)
Z (n)·S(n)= 1− · + i +O .
w 3ζ(4) p+p3 lnx lnix lnk+1x
n≤x p i=2
This proves Theorem 2.
Vol. 4 OnthePseudo-Smarandache-SquarefreefunctionandSmarandachefunction 11
References
[1]FeliceRusso,AsetofnewSmarandachefunctions,sequencesandconjecturesinnumber
theory, Lupton USA, American Research Press, 2000.
[2] Tom M Apostol. Introduction to Analytic Number Theory, New York: Spinger-Verlag,
1976.
[3]PanChengdongandPanChengbiao,Theelementaryproofoftheprimetheorem,Shang-
hai Science and Technology Press, Shanghai, 1988.