On the mean value of the Near Pseudo Smarandache Function

Scientia Magna Vol. 2 (2006), No. 2, 35-39

On the mean value of the Near Pseudo Smarandache Function Hai Yang1 and Ruiqin Fu2 1. Research Center for Basic Science, Xi’an Jiaotong University Xi’an, Shaanxi, P.R.China 2. School of Science, Department of Mathematics, Xi’an Shiyou University Xi’an, Shaanxi, P.R.China Abstract The main purpose of this paper is using the analytic method to study the asymptotic properties of the Near Pseudo Smarandache Function, and give two interesting asymptotic formulae for it.

Keywords

Near Pseudo Smarandache Function, mean value, asymptotic formula.

§1. Introduction In reference [1], David Gorski defined the Pseudo Smarandache function Z(n) as: let n be any positive integer, Z(n) is the smallest integer such that 1 + 2 + 3 + . . . + Z(n) is divisible by n. In reference [2], A.W.Vyawahare defined a new function K(n) which is a slight modification of Z(n) by adding a smallest natural number k, so this function is called “Near Pseudo Smarandache Function”. It is defined as follows: let n be any positive integer, K(n) = m, n P where m = n + k and k is the smallest natural number such that n divides m. About the n=1

mean value properties of the smallest natural number k in Near Pseudo Smarandache function, it seems that none had studied them before, at least we couldn’t find any reference about it.

In this paper, we use the analytic method to study the mean value properties of d(k) and ϕ(k), and give two interesting asymptotic formulae for it. That is, we shall prove the following: Theorem 1. Let k is the smallest natural number such that n divides Near Pseudo Smarandache function K(n), d(n) denotes Dirichlet divisor function. Then for any real number x ≥ 1, we have the asymptotic formula ¶ X X µ 1 n(n + 1) 3 d(k) = d K(n) − = x log x + Ax + O(x 2 log2 x), 2 4 n≤x

n≤x

where A is a computable constant.

Theorem 2. For any real number x ≥ 1, k is the smallest natural number such that n divides Near Pseudo Smarandache function K(n), ϕ(n) denotes the Euler’s totient function.

Then we have the asymptotic formula ¶ X µ 3 93 2 n(n + 1) x + O(x 2 +² ), = ϕ K(n) − 2 2 28π n≤x

where ² denotes any fixed positive number.

36

Hai Yang and Ruiqin Fu

No. 2

§2. Some lemmas To complete the proof of the theorems, we need the following several simple Lemmas: Lemma 1. Let n be any positive integer, then we have   n(n+3) , if n is odd, 2 K(n) =  n(n+2) , if n is even.

2 Proof. (See reference [2]).

Lemma 2. For any real number x ≥ 1, we have X ¡√ ¢ d(n) = x log x + (2C − 1)x + O x , n≤x

where C is the Euler constant, X

ϕ(k) =

n≤x

3 2 x + O(x log x). π2

Proof. These results can be get immediately from [3].

Lemma 3. For any real number x ≥ 1, we have µ ¶ ³ 1 ´ X log 2 3 3 − x + O x 2 log2 x , d(2n) = x log x + 2 2 2 n≤x

X

ϕ(2n) =

n≤x

3 2 x2 + O(x 2 +² ).

7ζ(2)

Proof. Firstly, we shall prove the first formula of Lemma 3. Let s = σ + it be a complex ∞ P d(2n) ² number and f (s) = ns . Note that d(2n) ¿ n , so it is clear that f (s) is a Dirichlet series n=1

absolutely convergent for Re(s)> 1, by the Euler product formula [3] and the definition of d(n) we get f (s) = =

∞ YX d(2pm ) pms p m=0 ∞ ∞ X d(2m+1 ) Y X d(2pm ) · 2ms pms m=0 p>2 m=0 Ã ! µ ∞ ∞ Q P P d(pm ) · pms

= 2ζ 2 (s) ·

p>2 m=0

m=0

¶

´ ∞ ³ Q P d(pm ) p m=0

∞ P

= 2ζ 2 (s) ·

d(2m+1 ) 2ms

m=0 ∞ P

pms

d(2m+1 ) 2ms

m=0

d(2m ) 2ms

µ ¶ 1 = ζ 2 (s) 2 − s .

2

(1)

Vol. 2

On the mean value of the Near Pseudo Smarandache Function

where ζ(s) is the Riemann zeta-function, and

Q

37

dentoes the product over all primes.

p

From (1) and the Perron’s formula [4], for b = 1 + ², T ≥ 1 and x ≥ 1 we have ¯ b¯ µ ¶ Z b+iT X ¯x ¯ 1 xs xH(2x) log x d(2n) = f (s) ds + O ¯¯ ¯¯ + O .

2πi b−iT s T T

(2)

n≤x

Taking a =

1 2

+ ², we move the integral line in (2). Then µ ¶ X 1 xs 2 d(2n) = Res ζ (s) 2 − s s=1 2 s n≤x ¯Z ¶ µ Z a+iT Z b+iT ¯¯ 1 ¯¯ a−iT 1 xs ¯ 2 ds + + ζ (s) 2 − + ¯ ¯ 2πi ¯ b−iT 2s s a−iT a+iT ¯ ¯ b¯ ¯ ¯ ¯x ¯ ¯ xH(2x) log x ¯ ¯, + O ¯¯ ¯¯ + O ¯¯ ¯ T T

where

¯Z ¯ ¯ a−iT Z b+iT ¯ 1 xs x ¯ ¯ 2 + ¯ ¯ ζ (s)(2 − s ) ds ¿ , ¯ b−iT 2 s T a+iT ¯ Z a+iT 1 1 xs ζ 2 (s)(2 − s ) ds ¿ x 2 log2 T. 2 s a−iT

Hence, we have X

¯x¯ 1 xs ¯ ¯ ) + O ¯ ¯ s=1 2s s T ¯ ¯ ¯ ¯ ³ 1 ´ ¯ ¯ xb ¯ log x ¯¯ 2 ¯ ¯ ¯ 2 + O x log T + O ¯ ¯ + O ¯xH(2x) T T ¯ ¯ ¯ s 1 x ¯x¯ = Res ζ 2 (s)(2 − s ) + O ¯ ¯ s=1 2 s¯ T ¯ ³ 1 ´ ¯ log x ¯¯ + O x 2 log2 T + O ¯¯x1+ε .

T ¯

d(2n)

=

n≤x

Res ζ 2 (s)(2 −

(3)

1

Taking T = x 2 +² in (3), then ³ 1 ´ ³ 1 ´ 1 xs 2 2 −ε 2 log x ) + O x + O x s=1 2s s n≤x µ ¶ ´ ³ 1 1 xs = Res ζ 2 (s) 2 − s + O x 2 log2 x .

(4) s=1 2 s ¡ ¢ s Now we can easily get the residue of the function ζ 2 (s) 2 − 21s · xs at second order pole point s = 1 with ¶ µ ¶ µ 3 log 2 3 1 xs = x log x + − x.

(5) Res ζ 2 (s) 2 − s s=1 2 s 2 2 2 X

d(2n)

= Res ζ 2 (s)(2 −

Combining (4) and (5), we may immediately get µ ¶ ´ ³ 1 X log 2 3 3 x + O x 2 log2 x . − d(2n) = x log x + 2 2 2 n≤x

38

Hai Yang and Ruiqin Fu

No. 2

This completes the proof of the first formula of Lemma 3. ∞ P ϕ(2n) Let h(s) = ns . From Euler product formula [2] and the definition of ϕ(n) we also have

n=1

! ∞ X ϕ(2pm ) h(s) = 1+ pms p m=1 Ã ! Ã ! ∞ ∞ X Y X ϕ(2m+1 ) ϕ(2pm ) = 1+ · 1+ 2ms pms m=1 p>2 m=1 ¶ µ ¶ µ ∞ ∞ P Q P ϕ(2m+1 ) ϕ(pm ) · 1 + 1+ ms ms p 2 ζ(s − 1) p>2 m=1 m=1 ¶ µ · = ∞ Q P ζ(s) ϕ(pm ) 1+ pms Y

Ã

p

µ

∞ P

m=1

m+1

¶

=

ϕ(2 ) 1+ 2ms ζ(s − 1) ¶ · µ m=1 ∞ P ζ(s) ϕ(2m ) 1+ ms 2

=

ζ(s − 1) 2s · s . ζ(s) 2 +3

m=1

By Perron formula [4] and the method of proving the first formula of Lemma 3, we can obtain the second formula of Lemma 3.

§3. Proof of the theorems In this section, we will complete the proof of the Theorems. From the first formula of Lemma 3 we can obtain µ ¶ ³ 1 ´ X 3 log 2 3 d(2n) = x log x − − x + O x 2 log2 x .

4 2 8 x n≤ 2

Let f (n) = K(n) −

n(n+1) 2

= k, then from Lemma 1 and the first formula of Lemma 2 we

have ¶ X µ n(n + 1) d(k) = d K(n) − 2 n≤x n≤x X n X = d( ) + d(n) 2 X

n≤x 2|n

=

X

n≤x 2†n

d(n) +

n≤ x 2

=

X

n≤x

d(n) −

X n≤ x 2

d(2n)

´ ³ 1 3 x log x + Ax + O x 2 log2 x , 4

where A is a computable constant.

This complets the proof of Theorem 1.

Vol. 2

On the mean value of the Near Pseudo Smarandache Function

Now we complete the proof of Theorem 2. Noting that ζ(2) = of Lemma 3 we can obtain X 3 3 2 ϕ(2n) = x + O(x 2 +² ).

2 7π x

π2 6 ,

39

from the second formula

n≤ 2

Then from Lemma 1 and the second formula of Lemma 2 we have ¶ X X µ n(n + 1) ϕ(k) = ϕ K(n) − n n≤x n≤x X X n ϕ(n) = ϕ( ) + 2 n≤x 2†n

n≤x 2|n

=

X

ϕ(n) +

n≤ x 2

=

X

n≤x

ϕ(n) −

³ 3 ´ 93 2 x + O x 2 +² , 2 28π

X

ϕ(2n)

n≤ x 2

where ² is any fixed positive number.

This complets the proof of Theorem 2.

References [1] David Gorski, The Pseudo Smarandache Function, Smarandache Notions Journal, 13(2002), 140-149.

[2] A.W.Vyawahare, Near Pseudo Smarandache Function, Smarandache Notions Journal, 14(2004), 42-61.

[3] Tom M.Apostol, Introduction to Analytic Number Theory, New York, Springer-Verlag, 1976.

[4] Pan Chengdong and Pan Chengbiao, Foundation of Analytic number Theory, Beijing, Science Press, 1997.