ON THE INTEGER PART OF THE K-TH ROOT OF A POSITIVE INTEGER

ON THE INTEGER PART OF THE K -TH ROOT OF A POSITIVE INTEGER ∗ Zhang Tianping 1. Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China 2. College of Mathematics and Information Science, Shannxi Normal University, Xi’an, Shaanxi, P.R.China tianpzhang@eyou.com

Ma Yuankui Department of Mathematics and Physics, Xi’an Institute of Technology, Xi’an, Shaanxi, P.R.China beijiguang2008@126.com

Abstract

For any positive integer £ m, let ¤ a(m) denotes the integer part of the k-th root of m. That is, a(m) = m1/k . In this paper, we study the asymptotic properties of σ−α (f (a(m))), X 1 where 0 < α ≤ 1 be a fixed real number, σ−α (n) = , f (x) be a lα l|n

polynomial with integer coefficients. An asymptotic formula is obtained.

Keywords:

Integer part sequence; k-th root; Mean value; Asymptotic formula.

§1. Introduction For any positive integer h m, let i a(m) denotes the integer part of the k-th root 1/k of m. That is, a(m) = m . For example, a(1) = 1, a(2) = 1, a(3) = 1, a(4) = 1, · · ·, a(2k ) = 2, a(2k + 1) = 2, · · ·, a(3k − 1) = 2, a(3k ) = 3, · · ·.

In problem 80 of reference [1], Professor F. Smarandach asked us to study the asymptotic properties of the sequence {a(m)}. About this problem, it seems that none had studied it, at least we have not seen related paper before. In this paper, we shall use the elementary method to study the asymptotic properties ∗ This

work is supported by the N.S.F.(60472068) and the P.N.S.F.of P.R.China .

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SCIENTIA MAGNA VOL.1, NO.1

of σ−α (f (a(m))), and give an interesting asymptotic formula. That is, we shall prove the following: Theorem. Let 0 < α ≤ 1 be a fixed real number, f (x) be a polynomial with integer coefficients. Then for any real number x > 1, we have the asymptotic formula ³ ´ X σ−α (f (a(m))) = Cf (α)x + O x1−α/k+ε , m≤x

where σ−α (n) =

X 1

l

l|n

, α

Cf (α) =

∞ X

Pf (d)d−1−α ,

X

Pf (d) =

d=1

f (n)≡0( mod d),0<n≤d

and ε denotes any fixed positive number.

§2. Several lemmas To complete the proof of the theorem, we need the following two simple lemmas: Lemma 1. Let 0 < α ≤ 1 be a fixed real number, f (x) be a polynomial with integer coefficients. Then for any real number x > 1, we have the asymptotic formula ³

X

´

σ−α (f (a(n))) = Cf (α)x + O x1−α lnγ x ,

n≤x,f (n)6=0

where γ is a certain constant, and Cf (α) =

∞ X

Pf (d)d−1−α ,

d=1

Pf (d) =

X

1.

f (n)≡0( mod d),0<n≤d

Proof. (See reference [2]).

Lemma 2. Let M be a fixed positive integer, f (x) be a polynomial with integer coefficients. Then we have M X

tk−1 σ−α (f (t)) =

t=1

Proof. Let A(y) =

X t≤y

[3]) we have M X t=1

tk−1 σ−α (f (t))

³ ´ Cf (α) k M + O M k−α lnγ M . k

σ−α (f (t)), by Abel’s identity (see Theorem 4.2 of

1,

On the integer part of the k -th root of a positive integer 1 Z

= M k−1 A(M ) − A(1) − (k − 1) ³

³

M

1 γ

y k−2 A(y)dy

= M k−1 Cf (α)M + O M 1−α ln M Z

−(k − 1)

1

³

M

´´ ³

y k−2 Cf (α)y + O y 1−α lnγ y

³

´

= Cf (α)M k + O M k−α lnγ M − =

7

³ ´ Cf (α) k M + O M k−α lnγ M . k

´´

dy

³ ´ Cf (α)(k − 1) k M + O M k−α lnγ M k

This completes the proof of Lemma 2.

§3. Proof of Theorem In this section, we shall complete the proof of Theorem. For any real number x ≥ 1, let M be a fixed positive integer such that M k ≤ x < (M + 1)k .

Let a0 denotes the constant term of f (x), from the definition of a(m) and Lemma 2, we have X

σ−α (f (a(m)))

m≤x

=

M X

X

t=1 (t−1)k ≤m<tk

=

M −1 X

X

σ−α (f (a(m))) +

X

=



´

X

M k ≤m≤(M +1)k

M −1 ³ X t=1

´

Ck1 tk−1 + Ck2 tk−2 + · · · + 1 σ−α (f (t)) 

+O 

 X

σ−α (f (M ))

M k ≤m≤(M +1)k

= k

M X t=1

σ−α (f (M ))

M k ≤m≤x

(t + 1)k − tk σ−α (f (t)) + O 

t=1

=

X

σ−α (f (t)) + σ−α (a0 ) +

t=1 tk ≤m<(t+1)k M −1 ³ X

σ−α (f (a(m)))

M k ≤m≤x

³

´

tk−1 σ−α (f (t)) + O M k−1 σ−α (f (t)) ³

´

= Cf (α)M k + O M k−α+ε , where we have used the fact that σ−α (n) ¿ nε .



σ−α (f (M ))

8

SCIENTIA MAGNA VOL.1, NO.1 On the other hand, note that the estimate 0 ≤ x − M k < (M + 1)k − M k = kM k−1 + Ck2 M k−2 + · · · + 1 µ ¶ k−1 1 k−1 2 1 = M k + Ck + k−1 ¿ x k .

M M

Now combining the above, we can immediately get the asymptotic formula X

³

´

σ−α (f (a(m))) = Cf (α)x + O x1−α/k+ε .

m≤x

This completes the proof of Theorem.

References [1] F. Smarandache, Only Problems, Not Solutions. Chicago: Xiquan Publ.

House, 1993.

[2] G. Babaev, N. Gafurov and D. Ismoliov, Some asymptotic formulas connected with divisors of polynomials, Trudy Mat. Inst. Steklov 163 (1984), 10-18.

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