On the mean value of the Smarandache ceil function

Scientia Magna Vol. 2 (2006), No. 1, 42-44

On the mean value of the1Smarandache ceil function Wang Xiaoying Research Center for Basic Science, Xi’an Jiaotong 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 properties of the Smarandache ceil function, and give a sharp asymptotic formula for it.

Keywords

Smarandache ceil function; Mean value; Asymptotic formula.

§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] [3] studied this function both theoretically and computationally, and got the following conclusions: (a, b) = 1 ⇒ Sk (ab) = Sk (a)Sk (b),

a, b ∈ N ∗ .

α1 αr αr 1 α2 Sk (pα 1 p2 · · · pr ) = Sk (p1 ) · · · Sk (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. For any real number x ≥ 2, we have the asymptotic formula X n≤x

1 1 3 ln2 x + A1 ln x + A2 + O(x− 4 +² ), = 2 S2 (n) 2π

where A1 and A2 are two computable constants, ² is any fixed positive integer.

1 This

work is supported by the N.S.F(10271093) and P.N.S.F of P.R.China

Vol. 2

On the mean value of the Smarandache ceil function

43

§2. Proof of the theorem To complete the proof of the theorem, we need the following Lemma, which is called the Perron’s formula (See reference [4]): ∞ X Lemma. Suppose that the Dirichlet series f (s) = a(n)n−s , s = σ + it, convergent n=1

absolutely for σ > σa , and that there exist a positive increasing function H(u) and a function B(u) such that a(n) ≤ H(n), and

∞ X

n = 1, 2, · · · ,

|a(n)| n−σ ≤ B(σ),

σ > σa .

n=1

Then for any s0 = σ0 + it0 , b0 > σa , b0 ≥ b > 0 , b0 ≥ σ0 + b > σa , T ≥ 1 and x ≥ 1, x not to be an integer, we have µ b ¶ xs x B(b + σ0 ) ds + O s T b−iT n≤x µ µ ¶¶ µ µ ¶¶ log x x 1−σ0 −σ0 + O x H(2x) min 1, +O x H(N ) min 1, , T T || x || X

a(n)n−s0 =

1 2πi

Z

b+iT

f (s0 + s)

where N is the nearest integer to x, || x ||= |N − x|.

Now we complete the proof of the theorem. Let s = σ + it be a complex number and f (s) =

∞ X

1 .

S (n)ns n=1 2

Note that | S21(n) | ≤ √1n , so it is clear that f (s) is a Dirichlet series absolutely convergent for Re(s) > 12 , by Euler product formula [5] and the definition of S2 (n) we have f (s)

=

Yµ p

1+

1 1 1 + + S2 (p)ps S2 (p2 )p2s S2 (p3 )p3s

¶ 1 1 1 + · · · + + + · · · S2 (p4 )p4s S2 (p2k )p2ks S2 (p2k+1 )p(2k+1)s µ Y 1 1 1 1 1 1 = 1 + s+1 + 2s+1 + 3s+2 + 4s+2 + · · · + 2ks+k + (2k+1)s+k+1 p p p p p p p ¶ 1 1 + 2(k+1)s+k+1 + (2(k+2)+1)s+k+2 + · · · p p ¶ µ Y 1 1 = 1 + 1 ps+1 1 − p2s+1 p +

=

ζ(2s + 1)ζ(s + 1) , ζ(2s + 2)

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

Y p

denotes the product over all primes.

44

Wang Xiaoying

Taking H(x) = 1;

B(σ) =

2 , 2σ − 1

No. 1

σ>

1 ; 2

5

s0 = 0; b = 1; T = x 4 in the above Lemma we may get X n≤x

1 1 = S2 (n) 2iπ

Z

5

1+ix 4

5 1−ix 4

f (s)

1 xs ds + O(x− 4 +ε ). s

5

To estimate the main term, we move the integral line in the above formula from s = 1±ix 4 s 5 to s = − 14 ± ix 4 . This time, the function f (s) xs have a third order pole point at s = 0 with residue 3 ln2 x + A1 ln x + A2 , 2π 2 where A1 and A2 are two computable constants.

Hence, we have   5 Z Z − 41 +ix 54 Z − 14 −ix 54 Z 1−ix 54 4 s 1  1+ix  ζ(2s + 1)ζ(s + 1)x ds + + + 5 5 5 5 2πi ζ(2s + 2)s 1−ix 4 1+ix 4 − 14 +ix 4 − 14 −ix 4 =

3 ln2 x + A1 ln x + A2 .

2π 2

We can easily get the estimate ¯ ¯   ¯ ¯ Z − 14 +ix 54 Z − 41 −i 45 Z 1−ix 54 s ¯ 1 ¯ 1 ζ(2s + 1)ζ(s + 1)x ¯   + + ds¯¯ ¿ x− 4 +² . ¯ 2πi 5 5 5 1 1 ζ(2s + 2)s 1+ix 4 − 4 −ix 4 − 4 +i 4 ¯ ¯ From above we may immediately get the asymptotic formula: X n≤x

1 3 1 = ln2 x + A1 ln x + A2 + O(x− 4 +² ).

S2 (n) 2π 2

This completes the proof of the theorem.

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, 1997.

[3] Ibstedt, Computational Aspects of Number Sequences, American Research Press, Lupton, 1999.

[4] Pan Chengdong and Pan Chengbiao, Goldbach conjecture, Science Press, Beijing, 1992, pp. 145.

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