The main purpose of this paper is using the elementary and analytic methods to study the value distribution properties of SDF(n), and give an interesting mean value formula for it

Wang, Jianping

English

ZENODO

Scientia MagnaVol. 3 (2007), No. 4, 111-114On the value distribution properties of theSmarandache double-factorial functionJianping WangCollege of Science, Chang’an University, Xi’an, 710064, P.R.ChinaAbstract For any positive integer n, the famous Smarandache double-factorial functionSDF (n) is defined as the smallest positive integer m, such that m!! is divisible by n, wherethe double factorial m!! = 1 · 3 · 5 · · ·m, if m is odd; and m!! = 2 · 4 · 6 · · ·m, if m is even. Themain purpose of this paper is using the elementary and analytic methods to study the valuedistribution properties of SDF (n), and give an interesting mean value formula for it.Keywords The Smarandache double-factorial function, value distribution, mean value, as-ymptotic formula.§1. Introduction and resultsFor any positive integer n, the famous Smarandache double-factorial function SDF (n) isdefined as the smallest positive integer m, such that m!! is divisible by n, where the doublefactorialm!! ={1 · 3 · 5 · · ·m, if m is odd ;2 · 4 · 6 · · ·m, if m is even.For example, the first few values of SDf(n) are:SDF (1) = 1, SDF (2) = 2, SDF (3) = 3, SDF (4) = 4, SDF (5) = 5, SDF (6) = 6,SDF (7) = 7, SDF (8) = 4, SDF (9) = 9, SDF (10) = 10, SDF (11) = 11, SDF (12) = 6,SDF (13) = 13, SDF (14) = 14, SDF (15) = 5, SDF (16) = 6 · · · · · · .In reference [1] and [2], F.Smarancdache asked us to study the properties of SDF (n).About this problem, some authors had studied it, and obtained some interesting results, seereference [3]. In an unpublished paper, Zhu Minhui proved that for any real number x > 1 andfixed positive integer k, we have the asymptotic formula∑n≤xSDF (n) =5pi248· x2lnx+k∑i=2ai · x2lni x+O(x2lnk+1 x),where ai are computable constants.112 Jianping Wang No. 4The other contents related to the Smarandache double-factorial function can also be foundin references [4], [5], [6] and [7]. For example, Dr. Xu Zhefeng [4] studied the value distributionproblem of the F.Smarandache function S(n), and proved the following conclusion:Let P (n) denotes the largest prime factor of n, then for any real number x > 1, we havethe asymptotic formula∑n≤x(S(n)− P (n))2 = 2ζ(32)x323 lnx+O(x32ln2 x),where ζ(s) denotes the Riemann zeta-function.The main purpose of this paper is using the elementary and analytic methods to studythe value distribution problem of the double-factorial function SDF (n), and give an interestingasymptotic formula it. That is, we shall prove the following conclusion:Theorem 1. For any real number x > 1 and any fixed positive integer k, we have theasymptotic formula∑n≤x(SDF (n)− P (n))2 = ζ(3)24x3lnx+k∑i=2ci · x3lni x+O(x3lnk+1 x),where P (n) denotes the largest prime divisor of n, and all ci are computable constants.Now we define another function S(n) as follows: Let S(n) denotes the smallest positiveinteger m such that n | m!. That is, S(n) = min{m : n|m!}. It is called the F.Smarandachefunction. For this function, using the method of proving Theorem 1 we can also get the following:Theorem 2. For any real number x > 1 any fixed positive integer k, we have the asymptoticformula∑n≤x(SDF (n)− S(n))2 = ζ(3)24x3lnx+k∑i=2ci · x3lni x+O(x3lnk+1 x).§2. Proof of the theoremsIn this section, we shall prove our theorems directly. First we prove Theorem 1. Weseparate all integers n in the interval [1, x] into two subsets A and B as follows: A = {n : 1 ≤n ≤ x, P (n) > √n}; B = {n : 1 ≤ n ≤ x, n /∈ A}, where P (n) denotes the largest primedivisor of n. If n ∈ A, then n = m · P (n) and P (m) < P (n). So from the definition of A wehave SDF (2) = 2. For any positive integer n > 2 and n ∈ A, SDF (n) = P (n), if 2 † n.Vol. 3 On the value distribution properties of the Smarandache double-factorial function 113SDF (n) = 2P (n), if 2 | n. From this properties we have∑n≤xn∈A(SDF (n)− P (n))2=∑2n≤x2n∈A(SDF (2n)− P (2n))2 +∑2n−1≤x2n−1∈A(SDF (2n− 1)− P (2n− 1))2=∑n≤ x22n∈A(SDF (2n)− P (2n))2 =∑1<n≤ x22n∈A(2P (2n)− P (2n))2=∑1<n≤ x22n∈AP 2(2n) =∑np≤ x2p>2np2 =∑n≤√x2∑2n<p≤ x2np2. (1)By the Abel’s summation formula (See Theorem 4.2 of [8]) and the Prime Theorem (See The-orem 3.2 of [9]):pi(x) =k∑i=1ai · xlni x+O(xlnk+1 x),where ai (i = 1, 2, · · · , k) are constants and a1 = 1.We have ∑2n<p≤ x2np2 =x2(2n)2· pi( x2n)− (2n)2 · pi(2n)− 2∫ x2n2ny · pi(y)dy=x324n3 lnx+k∑i=2bi · x3 · lni nn3 · lni x +O(x3n3 · lnk+1 x), (2)where we have used the estimate 2n ≤ √x, and all bi are computable constants.Note that∞∑n=11n3= ζ(3), from (1) and (2) we have∑n≤xn∈A(SDF (n)− P (n))2 = ζ(3)24x3lnx+k∑i=2ci · x3lni x+O(x3lnk+1 x), (3)where all ci are computable constants.For any positive integer n with n ∈ B, it is clear that SDF (n)¿ √n · lnn and P (n)¿ √n.So we have the estimate∑n≤xn∈B(SDF (n)− P (n))2 ¿∑n≤xn · ln2 n¿ x2 · ln2 x. (4)Combining (3) and (4) we have∑n≤x(SDF (n)− P (n))2 =∑n≤xn∈A(SDF (n)− P (n))2 +∑n≤xn∈B(SDF (n)− P (n))2=ζ(3)24x3lnx+k∑i=2ci · x3lni x+O(x3lnk+1 x),114 Jianping Wang No. 4where all ci are computable constants. This proves Theorem 1.Now we prove Theorem 2. Note that S(n)−P (n) = 0, if n ∈ A; and |S(n)−P (n)| ¿ √n,if n ∈ B. So from the result of the reference [4] and the proving method of Theorem 1 we have∑n≤x(SDF (n)− S(n))2 =∑n≤x(SDF (n)− P (n))2 +∑n≤x(S(n)− P (n))2−2∑n≤x(S(n)− P (n)) · (SDF (n)− S(n))=ζ(3)24x3lnx+k∑i=2ci · x3lni x+O(x3lnk+1 x).This completes the proof of Theorem 2.References[1] F. Smarandache, Only Problems, Not Solutions, Chicago, Xiquan Publishing House,1993.[2] M. L. Perez, Florentin Smarandache Definitions, Solved and Unsolved Problems, Con-jectures and Theorems in Number theory and Geometry, Chicago, Xiquan Publishing House,2000.[3] C. Dumitrescu and V. Seleccu, Some Notions and Questions in Number Theory, ErhusUniversity Press, Gelndale, 1994.[4] Xu Zhefeng, The value distribution property of the Smarandache function, Acta Math-ematica Sinica, Chinese Series, 49(2006), No. 5, 1009-1012.[5] Jianbin Chen, Value distribution of the F.Smarandache LCM function, Scientia Magna,3(2007), No. 2, 15-18.[6] Jozsef Sandor, On an generalization of the Smarandache function, Notes Numb. Th.Discr. Math, 5(1999), 41-51.[7] Jozsef Sandor, On an additive analogue of the function S, Smarandance Notes Numb.Th. Discr. Mathe, 5(2002), 266-270.[8] Tom M. Apostol, Introduction to analytical number theory, Spring-Verlag, New York,1976.[9] Pan Chengdong and Pan Chengbiao, The elementary proof of the prime theorem, Shang-hai Science and Technology Press, Shanghai, 1988.[10] Zhang Wenpeng, The elementary number theory, Shaanxi Normal University Press,Xi’an, 2007.

Florentin Smarandache Definitions, Solved and Unsolved Problems, Conjectures and Theorems

Introduction to analytical number theory,

On an additive analogue of the function S,

On an generalization of the Smarandache function,

Only Problems, Not Solutions,

Some Notions and Questions in Number Theory,

The elementary number theory,

The elementary proof of the prime theorem,

The value distribution property of the Smarandache function,

Value distribution of the F.Smarandache LCM function,

On the value distribution properties of the Smarandache double-factorial function

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

https://zenodo.org/record/9040/files/ValueDistribProp.pdf

On the value distribution properties of the Smarandache double-factorial function

Abstract

Similar works

Full text

Available Versions

ZENODO

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY