A new additive function and the F. Smarandache function

2009·0.12 MB·

by  Yanchun Guo

Scientia Magna
Vol. 5 (2009), No. 1, 128-132
A new additive function and
the F. Smarandache function
Yanchun Guo
Department of Mathematics, Xianyang Normal University
Xianyang, Shaanxi, P.R.China
Abstract For any positive integer n, we define the arithmetical function F(n) as F(1)=0.
If n > 1 and n = pα1pα2···pαk be the prime power factorization of n, then F(n) = α p +
1 2 k 1 1
α p +···+α p . Let S(n) be the Smarandache function. The main purpose of this paper
2 2 k k
is using the elementary method and the prime distribution theory to study the mean value
properties of (F(n)−S(n))2, and give a sharper asymptotic formula for it.
Keywords Additive function, Smarandache function, Mean square value, Elementary method,
Asymptotic formula.
§1. Introduction and result
Letf(n)beanarithmeticalfunction,wecallf(n)asanadditivefunction,ifforanypositive
integers m, n with (m, n) = 1, we have f(mn) = f(m)+f(n). We call f(n) as a complete
additive function, if for any positive integers r and s, f(rs) = f(r) + f(s). In elementary
number theory, there are many arithmetical functions satisfying the additive properties. For
example,ifn=pα1pα2···pαk denotestheprimepowerfactorizationofn,thenfunctionΩ(n)=
1 2 k
α +α +···+α and logarithmic function f(n) = lnn are two complete additive functions,
1 2 k
ω(n) = k is an additive function, but not a complete additive function. About the properties
of the additive functions, one can find them in references [1], [2] and [5].
Inthispaper,wedefineanewadditivefunctionF(n)asfollows: F(1)=0;Ifn>1andn=
pα1pα2···pαk denotestheprimepowerfactorizationofn,thenF(n)=α p +α p +···+α p .
1 2 k 1 1 2 2 k k
It is clear that this function is a complete additive function. In fact if m = pα1pα2···pαk
1 2 k
and n = pβ1pβ2···pβk, then we have mn = pα1+β1pα2+β2···pαk+βk. Therefore, F(mn) =
1 2 k 1 2 k
(α +β )p +(α +β )p +···+(α +β )p =F(m)+F(n). So F(n) is a complete additive
1 1 1 2 2 2 k k k
function. Now we let S(n) be the Smarandache function. That is, S(n) denotes the smallest
positive integer m such that n divide m!, or S(n) = min{m : n | m!}. About the properties
of S(n), many authors had studied it, and obtained a series results, see references [7], [8] and
[9]. The main purpose ofthis paper is usingthe elementarymethod and theprime distribution
theory to study the mean value properties of (F(n)−S(n))2, and give a sharper asymptotic
formula for it. That is, we shall prove the following:
Theorem. Let N be any fixed positive integer. Then for any real number x > 1, we
Vol. 5 AnewadditivefunctionandtheF.Smarandachefunction 129
have the asymptotic formula
(cid:181) (cid:182)
(cid:88) (cid:88)N x2 x2
(F(n)−S(n))2 = ci· lni+1x +O lnN+2√x ,
n≤x i=1
where c (i=1, 2, ··· , N) are computable constants, and c = π2.
i 1 6
§2. Proof of the theorem
Inthissection,weusetheelementarymethodandtheprimedistributiontheorytocomplete
the proof of the theorem. We using the idea in reference [4]. First we define four sets A, B,
C, D as follows: A = {n, n ∈ N, n has only one prime divisor p such that p | n and p2 (cid:45) n,
p > n13}; B = {n, n ∈ N, n has only one prime divisor p such that p2 | n and p > n13};
C ={n, n∈N, nhastwodeferentprimedivisorsp1 andp2 suchthatp1p2 |n, p2 >p1 >n31};
D = {n, n ∈ N, any prime divisor p of n satisfying p ≤ n13}, where N denotes the set of all
positive integers. It is clear that from the definitions of A, B, C and D we have
(cid:88) (cid:88) (cid:88)
(F(n)−S(n))2 = (F(n)−S(n))2+ (F(n)−S(n))2
n≤x n≤x n≤x
n∈A n∈B
(cid:88) (cid:88)
+ (F(n)−S(n))2+ (F(n)−S(n))2
n≤x n≤x
n∈C n∈D
≡ W +W +W +W . (1)
1 2 3 4
Now we estimate W , W , W and W in (1) respectively. Note that F(n) is a complete
1 2 3 4
additive function, and if n∈A with n=pk, then S(n)=S(p)=p, and any prime divisor q of
k satisfying q ≤ n13, so F(k) ≤ n31 lnn. From the Prime Theorem (See Chapter 3, Theorem 2
of [3]) we know that
(cid:181) (cid:182)
(cid:88) (cid:88)k x x
π(x)= 1= c · +O , (2)
i lnix lnk+1x
p≤x i=1
where c (i = 1, 2,··· , k) are computable constants, and c = 1. By these we have the
i 1
estimate:
(cid:88) (cid:88)
W = (F(n)−S(n))2 = (F(pk)−p)2
1
n≤x pk≤x
n∈A (pk)∈A
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
= F2(k)(cid:191) (pk)23 ln2(pk)≤(lnx)2 k23 p32
√ √
pk≤x k≤ xk<p≤x k≤ x k<p≤x
k k
(pk)∈A
(cid:88) (cid:179)x(cid:180)5 1
(cid:191) (lnx)2 k23 3 (cid:191)x53 ln2x. (3)
k lnx
k≤√x k
130 YanchunGuo No. 1
If n∈B, then n=p2k, and note that S(n)=S(p2)=2p, we have the estimate
(cid:88) (cid:88) (cid:161) (cid:162)
W = (F(n)−S(n))2 = F(p2k)−2p 2
2
n≤x p2k≤x
n∈B p>k
(cid:88) (cid:88) (cid:88) (cid:88)
= F2(k)(cid:191) k2
√ √
k≤x13 k<p≤ xk k≤x13 k<p≤ xk
(cid:88) k2·x12 x43
(cid:191) (cid:191) . (4)
k12 lnx lnx
1
k≤x3
If n∈D, then F(n)≤n13 lnn and S(n)≤n13 lnn, so we have
(cid:88) (cid:88)
W4 = (F(n)−S(n))2 (cid:191) n23 ln2n(cid:191)x35 ln2x. (5)
n≤x n≤x
n∈D
Finally, we estimate main term W3. Note that n ∈ C, n = p1p2k, p2 > p1 > n13 > k. If
k < p1 < n13, then in this case, the estimate is exact same as in the estimate of W1. If
k <p1 <p2 <n13, in this case, the estimate is exact same as in the estimate of W4. So by (2)
we have
(cid:88) (cid:88) (cid:179) (cid:180)
W3 = (F(n)−S(n))2 = (F(p1p2k)−p2)2+O x35 ln2x
n≤x p1p2k≤x
n(cid:88)∈C (cid:88) (cid:88) (cid:161)p2>p1>k (cid:162) (cid:179) (cid:180)
= F2(k)+2p1F(k)+p21 +O x53 ln2x
√
k≤x31 k<p1≤ xkp2≤p1xk  
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:179) (cid:180)
= p21+O kp1+O x35 ln2x
√ √
k≤x31 k<p1≤ xkp1<p2≤p1xk k≤x13 k<p1≤ xkp1<p2≤p1xk
(cid:88) (cid:88) (cid:195)(cid:88)N x (cid:181) x (cid:182)(cid:33) (cid:179) (cid:180)
= √ p21 ci· p klni x +O p klnN+1x +O x53 ln2x
k≤x13 k<p1≤ xk i=1 1 p1k 1 
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
 
− p21 1+O kp1. (6)
√ √
k≤x31 k<p1≤ xk p2≤p1 k≤x13 k<p1≤ xkp1<p2≤p1xk
Note that ζ(2)= π2, from the Abel’s identity (See Theorem 4.2 of [6]) and (2) we have
6
(cid:34) (cid:181) (cid:182)(cid:35)
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88)N c ·p p
p2 1= p2 i 1 +O 1
√ 1 √ 1 lnip lnN+1p
k≤x13 k<p1≤ xk p≤p1 k≤x13 k<p1≤ xk i=1 1  1
= (cid:88)N (cid:88) (cid:88) ci·p31 +O (cid:88) (cid:88) p31 
√ lnip √ lnN+1p
i=1k≤x31 k<p1≤ xk 1 k≤x13 k<p1≤ xk 1
(cid:181) (cid:182)
(cid:88)N d ·x2 2N ·x2
= i +O , (7)
lni+1x lnN+2x
i=1
Vol. 5 AnewadditivefunctionandtheF.Smarandachefunction 131
where d (i=1, 2, ··· , N) are computable constants, and d = π2.
i 1 6
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88) x (cid:88) x32 x53
kp (cid:191) k p · (cid:191) √ (cid:191) . (8)
k≤x13 k<p1≤√xkp1<p2≤p1xk 1 k≤x13 p1≤√xk 1 p1klnx k≤x13 kln2x ln2x
(cid:88) (cid:88) p x (cid:88) x2 x2
1 (cid:191) (cid:191) . (9)
√ klnN+1x k2lnN+2x lnN+2x
k≤x13 k<p1≤ xk k≤x13
From the Abel’s identity and (2) we also have the estimate
(cid:88) (cid:88) x (cid:88) 1 (cid:88) xp
p2 = 1
k≤x13 k<p1≤√xk 1p1klnpx1k k≤x13 k k<p1≤√xk lnkxp1
(cid:181) (cid:182)
(cid:88)N x2 x2
= b · +O , (10)
i lni+1x lnN+1x
i=1
where b (i=1, 2, ··· , N) are computable constants, and b = π2.
i 1 3
Now combining (1), (3), (4), (5), (6), (7), (8)and(9) we may immediately deduce the
asymptotic formula:
(cid:181) (cid:182)
(cid:88) (cid:88)N x2 x2
(F(n)−S(n))2 = ai· lni+1x +O lnN+2√x ,
n≤x i=1
where a (i=1, 2, ··· , N) are computable constants, and a =b −d = π2.
i 1 1 1 6
This completes the proof of Theorem.
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