Math Paper

ON THE MEAN VALUE OF SMARANDACHE CEIL FUNCTION

Ding Liping Department of Mathematics, Northwest University Xi’an, Shaanxi, P.R.China Abstract. For any fixed positive integer n, the Smarandache ceil function of order k is denoted by N ∗ → N and has the following definition: Sk (n) = min{x ∈ N | n | xk } (∀n ∈ N ∗ ) . In this paper, we study the mean value of properties the Smarandache ceil function, and give a sharp asymptotic formula for it.

1. Introduction For any fixed positive integer n, the Smarandache ceil function of order k is denoted by N ∗ → N and has the following definition: Sk (n) = min{x ∈ N | n | xk } (∀n ∈ N ∗ ) . For example, S2 (1) = 1, S2 (2) = 2, S2 (3) = 3, S2 (4) = 2, S2 (5) = 5, S2 (6) = 6, S2 (7) = 7, S2 (8) = 4, S2 (9) = 3, · · · . This was introduced by Smarandache who proposed many problems in [1]. There are many papers on the Smarandache ceil function. For example, Ibstedt [2] and [3] studied this function both theoretically and computationally, and got the following conclusions: (∀a, b ∈ N ∗ )(a, b) = 1 ⇒ Sk (ab) = Sk (a)Sk (b), α1 αr αr 1 α2 Sk (pα 1 p2 . · · · .pr ) = S( p1 ). · · · .S( pr ).

In this paper, we study the mean value properties of the Smarandache ceil function, and give a sharp asymptotic formula for it. That is, we shall prove the following: Theorem. Let x ≥ 2, for any fixed positive integer k, we have the asymptotic formula X n≤x

µ ¶¸ ³ 3 ´ Y· 1 1 x2 1− ζ(2k − 1) 1 + 2k−3 + O x 2 +² . Sk (n) = 2 p(p + 1) p p

Key words and phrases. Smarandache ceil function; Mean value; Asymptotic formula.. *This work is supported by the N.S.F.(10271093) and P.N.S.F of P.R.China Typeset by AMS-TEX

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DING LIPING

where ζ(s) is Riemann zeta function,

Y

denotes the product over all prime p, and

p

² be any fixed positive integer. This solved a conjecture of [4]. From this theorem we may immediately deduce the following: Corollary. For any real number x ≥ 2, we have the asymptotic formula: X n≤x

S2 (n) =

³ 3 ´ 3x2 ζ(3) + O x 2 +² . π2

2. Some Lemmas To complete the proof of the theorem, we need the following famous Perron formula [5]: Lemma. Suppose that the Dirichlet series f (s) =

∞ X

a(n)n−s , s = σ+it, converge

n=1

absolutely for σ > β, and that there exist a positive λ and a positive increasing function A(s) such that ∞ X

|a(n)| n−σ ¿ (σ − β)−1 , σ → β + 0

n=1

and a(n) ¿ A(n), n = 1, 2, · · · . Then for any b > 0, b + σ > β, and x not to be an integer, we have X

a(n)n

−s0

n≤x

1 = 2πi

Z

b+iT

b−iT

xω f (s0 +ω) dω+O ω

µ

¶ µ ¶ xb A(2x)x1−σ log x +O , T (b + σ − β)λ T || x ||

where || x || is the nearest integer to x. 3. Proof of the Theorem In this section, we complete the proof of Theorem. Let

f (s) =

Re(s) > 3.

∞ X Sk (n) , s n n=1

ON THE MEAN VALUE OF SMARANDACHE CEIL FUNCTION

By Euler product formula [6], we have ¶ Yµ Sk (p) Sk (p2 ) Sk (pk ) f (s) = 1+ + + ··· + + ··· s 2s ks p p p p ¶ Yµ p2 p p p p2 = 1 + s + 2s + · · · + ks + (k+1)s + · · · + 2ks + · · · p p p p p p ! à 1 1 Y p2 1 − pks 1 1 − pks + (k+1)s + ··· = 1 + s−1 p p 1 − p1s 1 − p1s p à ! 1 Y 1 − p1ks s−1 p = 1+ 1 1 1 − 1 − ps pks−1 p à µ ¶! ζ(s)ζ(s − 1)ζ(ks − 1) Y 1 1 1 1− = + s 1 ζ(2s − 2) pks−1 p 1 + ps−1 p where ζ(s) is Riemann zeta function. 5 Taking s0 = 0, b = 3, T = x 2 in the Lemma, we have Z 3+iT X 3 xs 1 ζ(s)ζ(s − 1)ζ(ks − 1) R(s) ds + O(x 2 +ε ), Sk (n) = 2iπ 3−iT ζ(2s − 2) s n≤x

where R(s) =

Y p

à 1−

µ

1 1+

1

ps−1

1 pks−1

+

1 ps

¶! .

To estimate the main term Z 3+iT 1 ζ(s)ζ(s − 1)ζ(ks − 1) xs R(s) ds, 2iπ 3−iT ζ(2s − 2) s we move the integral line from s = 3 ± iT to s = f (s) =

3 2

± iT . This time, the function

ζ(s)ζ(s − 1)ζ(ks − 1)xs R(s) ζ(2s − 2)s 2

has a simple pole point at s = 2 with residue x2 ζ(2k − 1)R(2). So we have ÃZ Z 32 +iT Z 32 −iT Z 3−iT ! 3+iT 1 ζ(s)ζ(s − 1)ζ(ks − 1)xs + + + R(s)ds 3 3 2iπ ζ(2s − 2)s 3−iT 3+iT 2 +iT 2 −iT µ ¶¸ Y· 1 x2 1 = ζ(2k − 1) 1− 1 + 2k−3 . 2 p(p + 1) p p Note that ÃZ 3 Z 32 −iT Z 3−iT ! 2 +iT 3 ζ(s)ζ(s − 1)ζ(ks − 1)xs 1 + + R(s)ds ¿ x 2 +² 3 3 2iπ ζ(2s − 2)s 3+iT 2 +iT 2 −iT From above we immediately get the asymptotic formula: µ ¶¸ ³ 3 ´ X Y· 1 x2 1 1− Sk (n) = ζ(2k − 1) 1 + 2k−3 + O x 2 +² . 2 p(p + 1) p p n≤x

This completes the proof of Theorem.

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DING LIPING

Acknowledgments The author express her gratitude to her supervisor Professor Zhang Wenpeng for his very helpful and detailed instructions. References 1. F. Smarandache, Only problems, not Solutions, Xiquan Publ. House, Chicago, 1993,pp. 42. 2. Ibstedt, Surfing on the Ocean of Numbers-a few Smarandache Notions and Similar TOPICS, Erhus University press, New Mexico,USA., 1997. 3. Ibstedt, Computational Aspects of Number Sequences, American Research Press, Lupton USA, 1999. 4. S. Tabirca and T. Tabirca, Smarandache notions journal 13 (2002), 30-36. 5. Pan Chengdong and Pan Chengbiao, Goldbach conjecture, Science Press, Beijing, 1992, pp. 145. 6. Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.