Mean value of a Smarandache-Type Function

Scientia Magna Vol. 2 (2006), No. 2, 31-34

Mean value of a Smarandache-Type Function Jia Wang Department of Mathematics, Shandong Normal University Jinan, Shandong, P.R.China Abstract In this paper, we use analytic method to study the mean value properties of Smarandache-Type Multiplicative Functions Km (n), and give its asymptotic formula . Finally, the convolution method is used to improve the error term.

Keywords

Smarandache-Type Multiplicative Function, the Convolution method.

§1. Introduction αk 1 α2 Suppose m ≥ 2 is a fixed positive integer. If n = pα 1 p2 ...pk , we define

Km (n) = pβ1 1 pβ2 2 ...pβkk ,

βi = min(αi , m − 1),

which is a Smarandache-type multiplicative function . Yang Cundian and Li Chao proved in [1] that ! Ã X 3 x2 Y 1 Km (n) = + O(x 2 +² ).

1+ m 2ζ(m) p (p − 1)(p + 1) n≤x

In this paper, we shall use the convolution method to prove the following Theorem.

The asymptotic formula ! Ã X 1 x2 Y 1 Km (n) = + O(x1+ m e−c0 δ(x) ) 1+ m 2ζ(m) p (p − 1)(p + 1)

n≤x

holds, where c0 is an absolute positive constant and δ(x) = (log x)3/5 (log log x)−1/5 .

§2. Proof of the theorem In order to prove our Theorem, we need the following Lemma, which is Lemma 14.2 of [2].

Lemma. Let f (n) be an arithmetical function for which : X

f (n) =

l X

xaj Pj (log x) + O(xa ),

j=1

n≤x

X n≤x

| f (n) |= O(xa1 logr x),

32

Jia Wang

No. 2

where a1 ≥ a2 ≥ . . . ≥ al > 1/k > a ≥ 0, r ≥ 0, P1 (t), · · · , Pl (t) are polynomials in t of degrees not exceeding r, and k ≥ 1 is a fixed integer. If h(n) =

X

µ(d)f (n/dk ),

dk |n

then X

h(n) =

l X

xaj Rj (log x) + E(x),

j=1

n≤x

where R1 (t), . . . , Rl (t) are polynomials in t of degrees not exceeding r, and for some D>0 ³ ´ E(x) ¿ x1/k exp − D(log x)3/5 (log log x)−1/5 .

Now we prove our Theorem. Let

∞ X Km (n) , <(s) > 2. ns n=1

g(s) =

According to Euler’s product formula, we write ! à Y Km (p) Km (p2 ) + + ··· g(s) = 1+ ps p2s p à !

Y Km (p) (Km (p2 )) = 1+ + + ··· ps p2s p à !

Y p p2 pm−1 pm−1 pm−1 = 1 + s + 2s + · · · + (m−1)s + ms + (m+1)s + · · · p p p p p p à !

Y 1 pm−1 1 1 pm−1 = 1 + s−1 + 2(s−1) + · · · + (m−1)(s−1) + ms + (m+1)s + · · · p p p p p p à !

1 Y 1 − pm(s−1) pm−1 1 1 + (1 + s + 2s + · · · ) = 1 ms p p p 1 − ps−1 p ! à 1 Y 1 − pm(s−1) pm−1 1 = + 1 pms 1 − p1s 1 − ps−1 p à !

1 Y 1 − pm(s−1) ps−1 − 1 1+ s = 1 (p − 1)(pm(s−1) − 1) 1 − ps−1 p ζ(s − 1) ¢ R(s), = ¡ ζ m(s − 1) where R(s) =

Y p

Ã

! ps−1 − 1 1+ s . (p − 1)(pm(s−1) − 1)

Vol. 2

33

Mean value of a Smarandache-Type Function

Let qm (n) denote the characteristic function of m-free numbers, then ∞ X ζ(s) qm (n) = , ζ(ms) n=1 ns

∞ X ζ(s − 1) qm (n)n = . ζ(m(s − 1)) n=1 ns

Suppose R(s) = then Km (n) =

∞ X r(n) , ns n=1

X

qm (l1 )l1 r(l2 ).

n=l1 l2

Obviously, when σ > 1, R(s) absolutely converges, namely X | r(l) |¿ x1+ε .

(1)

l≤x

We can write qm (n) as the following form qm (n) =

X

µ(d)

dk |n

Now we apply the lemma on taking f (n) = 1, l = a1 = 1, r = a = 0, then we have µ ¶ X 1 x qm (n) = + O x m e−c1 δ(x) ζ(m) n≤x

for some absolute constant c1 > 0.

By partial summation, X

qm (n)n =

n≤x

1 x2 + O(x1+ m e−c2 δ(x) ) 2ζ(m)

(2)

holds for some absolute constant c2 > 0. Let y = x1−1/2m . By hyperbolic summation , we write X X Km (n) = qm (l1 )l1 r(l2 ) n≤x

l1 l2 ≤x

=

X

r(l2 )

l1 ≤ lx

l2 ≤y

=

X

1

X

+

2

X

2

−

qm (l1 )l1 +

X

qm (l1 )l1

l1 ≤ x y

X 3

X

r(l2 ) −

l1 ≤ lx

1

X l2 ≤y

r(l2 )

X

(3)

qm (l1 )l1

l1 ≤ x y

.

From (1) we get X 2

¿

µ ¶1+ε x ¿ l1 l 1 x

X

x2+ε y

¿ x1+1/2m+ε .

(4)

l1 ≤ y

Similarly X 3

¿

x2+ε ¿ x1+1/2m+ε . y

(5)

34 Finally for

Jia Wang

P 1

No. 2

we have by (2) X 1

if we noticed that

µX ¡ ¢¶ x 1 −1− 1 x2 X r(l2 ) 1+ m m −c2 δ l2 +O x e l2 = 2ζ(m) l22 l2 ≤y l2 ≤y µ X ¶ µ ¡ ¢¶ 1 x2 r(l2 ) −c0 δ x 1+ m = R(2) + O x2 + O x e 2ζ(m) l22 l2 >y µ 2+ε ¶ µ ¡ ¢¶ 1 x2 x = R(2) + O + O x1+ m e−c0 δ x 2ζ(m) y ¶ µ 2 1 x = R(2) + O x1+ m e−c0 δ(x) , 2ζ(m)

(6)

X r(l2 ) ¿ y −1+ε , l22

l2 >y

which follows from (1) by partial summation.

Now our Theorem follows from (3)-(6).

References [1] Cundian Yang and Chao Li, Asymptotion Formulae of Smarandache-Type Multiplicative Functions, Hexis, 2004, pp. 139-142.

[2] A. Ivi´c, The Riemann zeta-function, Wiley, New York, 1985.