ON THE SMARANDCHE FUNCTION AND ITS HYBRID MEAN VALUE Yao Weili Research Center for Basic Science, Xi’an Jiaotong University, Xi’an, Shaanxi, P.R.China
Abstract
For any positive integer n, let S(n) denotes the Smarandache function, then S(n) is defined as the smallest m ∈ N + with n|m!. In this paper, we study the asymptotic property of a hybrid mean value of the Smarandache function and the Mangoldt function, and give an interesting hybrid mean value formula for it.
Keywords:
the Smarandche function; the Mangoldt function; Mean value.
§1. Introduction For any positive integer n, let S(n) denotes the Smarandache function, then S(n) is defined as the smallest m ∈ N + with n|m!. From the definition of S(n), one can easily deduce that if n = pα1 1 pα2 2 · · · pαk k is the prime power factorization of n, then S(n) = max S(pαi i ). 1≤i≤k
About the arithmetical properties of S(n), many people had studied it before (see reference [2]). In this paper, we study the asymptotic property of a hybrid mean value of the Smarandache function and the Mangoldt function, and give an interesting hybrid mean value formula for it. That is, we shall prove the following: Theorem. For any real number x ≥ 1, we have the asymptotic formula X
x2 ∧(n)S(n) = +O 4 n≤x
Ã
!
x2 log log x , log x
where ∧(n) is the Mangoldt function defined by ½
∧(n) =
log p, if n = pα (α ≥ 1); 0, otherwise.
§2. Proof of the theorem In this section, we shall complete the proof of the theorem. Firstly, we need following:
80
SCIENTIA MAGNA VOL.1, NO.1 Lemma. For any prime p and any positive integer α, we have µ
S(pα ) = (p − 1)α + O
¶
p log α . log p
Proof. From Theorem 1.4 of reference [3], we can obtain the estimate. Now we use the above Lemma to complete the proof of the theorem. From the definition of ∧(n), we have X
∧(n)S(n)
n≤x
X
=
S(pα ) log p
pα ≤x
µ
X X
=
µ
log p (p − 1)α + O
p≤x α≤ log x log p
X
=
¶¶
X
(p − 1) log p
p≤x
p log α log p
X
α+O
p≤x
x α≤ log log p
X
p
log α .
x α≤ log log p
Applying Euler’s summation formula, we can get µ
¶
log x 1 log2 x +O α= , 2 2 log p log p log x
X
α≤ log p
and
X
µ
log α =
x α≤ log log p
¶
log x log x log x log x log − + O log . log p log p log p log p
Therefore we have
X p X p 1 . (1) + O log x log log x ∧(n)S(n) = log2 x 2 log p log p n≤x p≤x p≤x X
If x > 0 let π(x) denote the number of primes not exceeding x, and let ½
a(n) = then π(x) =
X
1, if n is a prime; 0, otherwise.
a(n). Note the asymptotic formula
p≤x
µ
¶
x x +O , π(x) = log x log2 x
81
On the Smarandche function and itshybrid mean value
and from Abel’s identity, we have X
p log p p≤x =
X
a(n)
n≤x
n log n µ
Z
¶
x x 2 t − π(2) − π(t)d log x log 2 log t 2 µ µ ¶¶ Z x µ µ ¶¶ µ ¶ x x x t t t +O − + O d log x log x log t log t log2 x log2 t 2
= π(x) = =
1 x2 +O 2 log2 x
Ã
!
x2 . log3 x
(2)
Combining (1) and (2), we have X
∧(n)S(n)
n≤x
=
1 2 x +O 4
=
1 2 x +O 4
à Ã
x2 log x
!
Ã
x2 + O log x log log x 2 log x
!
!
x2 log log x . log x
This completes the proof of the theorem.
References [1] F. Smarandache, Only Problems, Not Solutions, Xiquan Publishing House, Chicago, 1993. [2] Jozef Sandor, On an generalization of the Smarandache function, Notes Numb. Th. Discr. Math. 5 (1999), 41-51. [3] Mark Farris and Patrick Mitchell, Bounding the Smarandache Function, Smarandache Notions Journal 13 (2003), 37-42. [4] T. M. Apostol, Introduction to Analytic Number Theory, New York, Springer-Verlag, 1976.