On The Inferior And Superior

ON THE INFERIOR AND SUPERIOR k-TH POWER PART OF A POSITIVE INTEGER AND DIVISOR FUNCTION

ZHENG JIANFENG Shaanxi Financial & Economics Professional College, Xianyang, Shaanxi, P.R.China ABSTRACT: For any positive integer n, let a(n) and b(n) denote the inferior and superior k-th power part of n respectively. That is, a(n) denotes the largest k-th power less than or equal to n, and b(n) denotes the smallest k-th power greater than or equal to n. In this paper, we study the properties of the sequences {a(n)} and {b(n)}, and give two interesting asymptotic formulas.

Key words and phrases: Inferior and superior k-th power part; Mean value; Asymptotic formula. 1. INTRODUCTION For a fixed positive integer k>1, and any positive integer n, let a(n) and b(n) denote the inferior and superior k-th power part of n respectively .That is, a(n) denotes the largest k-th power less than or equal to n, b(n) denotes the smallest k-th power greater than or equal to n. For example, let k=2 then a(1)=a(2)=a(3)=1,a(4)=a(5)=…=a(7)=4, …,b(1)=1, b(2)=b(3)=b(4)=4, b(5)=b(6)= … =b(8)=8…; let k=3 then a(1)=a(2)= …=a(7)=1, a(8)=a(9)= …=a(26)=8,…,b(1)=1, b(2)=b(3)= … =b(8)=8, b(9)=b(10)=…=b(27)=27…. In problem 40 and 41 of [1], Professor F . Smarandache asks us to study the properties of the sequences {a(n)} and {b(n)}. About these problems, Professor Zhang Wenpeng [4] gave two interesting asymptotic formulas of the cure part of a positive integer. In this paper, we give asymptotic formulas of the k-th power part of a positive integer. That is, we shall prove the following: Theorem 1. For any real number x>1 , we have the asymptotic formula 1

1− + ε 1 6 ∑ d (a(n)) = kk! ( 2 ) k −1 A0 x ln k x + A1x ln k −1 x +  + Ak −1x ln x + Ak x + O( x 2k ) kπ n≤ x

. where A0, A1, … Ak are constants, especially when k equals to 2, A0=1; d(n) denotes the Dirichlet divisor function,εis any fixed positive number. For the sequence {b(n)} , we can also get similar result. Theorem 2. For any real number x>1, we have the asymptotic formula 1

1− +ε 1 6 ∑ d (b(n)) = kk! ( 2 )k −1 A0 x ln k x + A1x ln k −1 x +  + Ak −1x ln x + Ak x + O( x 2k ) kπ n≤ x

. 2. A SIMPLE LEMMA To complete the proof of the theorems, we need following Lemma 1. For any real number x>1, we have the asymptotic formula



1

∑ d (n k ) = n≤ x

1

+ε 1 6 k −1 ( 2 ) B0 x ln k x + B1 x ln k −1 x +  + Bk −1 x ln x + Bk x + O( x 2 ) . k! π

where B0, B1,… Bk are constants, especially when k=2, A0=1;εis any fixed positive number. ∞

Proof. Let s = σ + it be a complex number and f ( s ) = ∑ n =1

d (n k ) . ns

ε

Note that d (n ) << n , So it is clear that f(s) is a Dirichlet series absolutely convergent in k

Re(s)>1, by the Euler Product formula [2] and the definition of d(n) we have

  d ( pk ) d ( p2k ) d ( p kn ) f ( s ) = ∏  1 + p s + p 2 s + + p ns +  p    k + 1 2k + 1  kn + 1 = ∏ 1 + s + 2 s +  + ns +  p p p p    1   =ζ 2 ( s )∏ 1 +( k −1) p s   p    1 1 1  = ζ 2 ( s )∏  (1 + s ) k −1 − C k2−1 2 s −  − ( k −1) s  p p p p   =

ζ k +1 ( s ) g ( s) . ζ k −1 ( 2s )

(1)

where ζ (s ) is Riemann zeta-function and

∏ p

denotes the product over all primes.

From (1) and Perron’s formula [3] we have

∑ d (n k ) = n≤ x

 x 2 +ε 1 2+iT ζ k +1 ( s ) xs  g ( s ) ds + O 2πi ∫2−iT ζ k −1 (2 s) s  T

where g (s ) is absolutely convergent in Re(s ) >

Re(s ) =

  , 

(2)

1 + ε . We move the integration in (2) to 2

1 + ε . The pole at s = 1 contributes to 2

1 6 k −1 ( 2 ) B0 x ln k x + B1 x ln k −1 x +  + Bk −1 x ln x + Bk x , k! π

(3)

where B0 , B1 ,… Bk are constants, especially when k = 2, B0 = 1 . For

1−σ 1 ≤ σ < 1 , note that ζ ( s) = ζ (σ + it ) ≤ t 2 +ε . Thus, the horizontal integral contributes to 2

 1 +ε x 2  O x 2 +  , T   2

(4)

and the vertical integral contributes to

 1 +ε  O x 2 ln 4 T  .   On the line Re(s ) =

(5)

3 1 + ε , taking parameter T = x 2 , then combining (2), (3), (4) and (5) we have 2

1 6  d (n ) =  2  ∑ k!  π  n≤ x k

k −1

B0 x ln x + B1 x ln k

k −1

 12 + ε  x +  + Bk x + O x .  

This proves Lemma 1. 3. PROOFS OF THE THEOREMS Now we complete the proof of the Theorems. First we prove Theorem 1. For any real number x > 1 , Let M be a fixed positive integer such that

M k ≤ x < ( M + 1) k ,

(6)

then, from the definition of a(n), we have

∑ d (a(n)) =

n≤ x

=

=

M

∑

∑ d (a(n) + ∑ d (a(n))

m = 2 ( m −1) k ≤ n < m k M −1

∑

M k ≤ n≤ x

∑ d (m k ) + ∑ d ( M k )

m =1 m k ≤ n < ( m +1) k

M k ≤ n≤ x

M −1



m =1

 M k ≤ n ≤ ( M +1) k

∑ (Ck1 mk −1 + Ck2 m k − 2 +  + 1)d (m k ) + O



∑ d ( M k )  , 

M

= k ∑ m k −1d (m k ) + O( M k −1+ ε) ,

(7)

m =1

ε

where we have used the estimate d (n) << n . Let M

B( y) = ∑ d (n k )

∑m m =1

n≤ y

, then by Abel’s identity and Lemma 1, we have M

d (m k ) = M k −1B ( M ) − (k − 1) ∫ y k − 2 B( y )dy + O(1)

k −1

1

=M

k −1

 1 6 k −1  k k −1  ( 2 ) B0 M ln M + B1M ln M +  + Bk M   k! π 

M 1 6  − (k − 1) ∫  ( 2 ) k −1 B0 y k −1 ln k y + B1 y k −1 ln k −1 y +  + Bk y k −1 dy 1  k! π 

3

 k − 12 + ε   +O  M    1  6  =   kk!  π 2 

k −1

B0 M ln M + C1M ln k

k

k

k −1

 k − 12 + ε  . M +  + Ck −1M + O M    k

(8)

Applying (7) and (8) we obtain the asymptotic formula

∑ d (a(n)) = n≤ x

1 6    k! π 2 

k −1

 k − 1 +ε  B0 M k ln k M + C1M k ln k −1 M +  + Ck −1M k + O M 2  , (9)  

where B0 , C1 ,, Ck −1 are constants. From (6) we have the estimates

0 ≤ x − M k < ( M + 1) k − M k = kM k −1 + Ck2 M k − 2 +  + 1 = M k −1 (k + Ck2

1 1 +  + k −1 ) << x M M

k −1 k

,

(10)

and 1  k −1  − +ε ln x ln k x = k k ln k M + O 1  = k k ln k M + O( x k ) .  k   x 

(11)

Combining (9), (10) and (11) we have 1

1− + ε 1 6 ∑ d (a(n)) = kk! ( 2 ) k −1 A0 x ln k x + A1x ln k −1 x +  + Ak −1x ln x + Ak x + O( x 2k ) , kπ n≤ x

where A0 equals to B0 . This proves Theorem 1. Using the methods of proving Theorem 1 we can also prove Theorem 2. This completes the proof of the Theorems.

Acknowledgments The author expresses his gratitude to professor Zhang Wenpeng for his very helps and detailed instructions.

REFERENCES: 1. F.Smarandache, Only problems, not Solutions, Xiquan Publ. House, Chicago, 1993, PP. 35. 2. T.M.Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. 3. Pan Chengdong and Pan Chengbiao, Foundation of Analytic Number Theory, Science Press, Beijing, 1997, PP. 98. 4. Zhang Wenpeng, On the cube part sequence of a positive integer (to appear).

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