HYBRID MEAN VALUE ON SOME SMARANDACHETYPE MULTIPLICATIVE FUNCTIONS AND THE MANGOLDT FUNCTION∗ Liu Huaning Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China hnliu@nwu.edu.cn
Gao Jing School of Science, Xi’an Jiaotong University, Xi’an, Shaanxi, P.R.China
Abstract
In this paper, we study the hybrid mean value of some Smarandache-type multiplicative functions and the Mangoldt function, and give two asymptotic formulae.
Keywords:
Smarandache-type multiplicative functions; Mangoldt function; Hybrid mean value.
§1. Introduction In [1], Henry Bottomley considered eleven particular families of interrelated multiplicative functions, which are listed in Smarandache’s problems. It might be interesting to discuss the mean value of these functions on {pα }, since they are multiplicative. In this paper we study the hybrid mean value of some Smarandache-type multiplicative functions and the Mangoldt function. One is Cm (n), which is defined as the m-th root of largest m-th power dividing n. The other function Jm (n) is denoted as m-th root of smallest m-th power divisible by n. We will give two asymptotic formulae on these two functions. That is, we shall prove the following: Theorem 1. For any integer m ≥ 3 and real number x ≥ 1, we have X
µ
Λ(n)Cm (n) = x + O
n≤x
¶
x , log x
where Λ(n) is the Mangoldt function. ∗ This
work is supported by the N.S.F.(10271093) and the P.S.F. of P.R.China.
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SCIENTIA MAGNA VOL.1, NO.1
Theorem 2. For any integer m ≥ 2 and real number x ≥ 1, we have Ã
X
2
Λ(n)Jm (n) = x + O
n≤x
!
x2 . log x
Using our methods one should be able to get some similar mean value formulae. We are hoping to see more papers.
§2. Proof of the theorems Now we prove the theorems. Noting that Cm (pα ) = pk ,
if mk ≤ α < m(k + 1)
(1)
and α
Cm (pα ) ≤ p m , then we have X
X
Λ(n)Cm (n) =
log pCm (pα ) =
pα ≤x
n≤x
½
1, 0,
a(n) =
X
log pCm (p) +
log pCm (pα ). (3)
pα ≤x α≥2
p≤x
Let
then
X
(2)
if n is prime; otherwise, µ
X
¶
x x +O . a(n) = π(x) = log x log2 x n≤x
By Abel’s identity and (1) we have X
X
log pCm (p) =
p≤x
µ
=x+O
x log x
¶
p≤x
pα ≤x α≥2
¿
log pCm (pα ) =
X
Z
a(n) log n = π(x) log x −
n≤x x 1
µZ
+O
From (2) we also have X
log p
2
X
¶
log t
dt = x + O
X 1
x 2≤α≤ log log 2 p≤x α
X
1
1
1
µ
2
x
π(t) dt t
¶
x . log x
log pCm (pα ) ≤
X
(4) X
α
log p · p m
1
x 2≤α≤ log log 2 p≤x α
1
xm+α ¿ xm+2 .
x 2≤α≤ log log 2
Therefore for any integer m ≥ 3 and real number x ≥ 1, from (3), (4) and (5) we have µ ¶ X x Λ(n)Cm (n) = x + O . log x n≤x
(5)
Hybrid mean value on some Smarandache-type multiplicative functions and theMangoldt function1 151
This proves Theorem 1. On the other hand, noting that Jm (pα ) = pk+1 ,
if mk < α ≤ m(k + 1)
and α
Jm (pα ) ≤ p m +1 , then using the methods of proving Theorem 1 we can easily get Theorem 2.
References [1] Henry Bottomley, Some Smarandache-Type Multiplicative Functions, Smarandache Notions Journal, 13 (2002), 134-135.