For any positive integer n, the F.Smarandache LCM function SL(n) is defined
as the smallest positive integer k

Xiaoyan, Li

English

ZENODO

Scientia MagnaVol. 3 (2007), No. 3, 58-62On the mean value of the Smarandache LCMfunctionXiaoyan LiDepartment of Mathematics, Northwest UniversityXi’an, Shaanxi, P.R.ChinaAbstract For any positive integer n, the F.Smarandache LCM function SL(n) is definedas the smallest positive integer k such that n | [1, 2, · · · , k], where [1, 2, · · · , k] denotes theleast common multiple of 1, 2, · · · , k. The main purpose of this paper is using the elementarymethods to study the mean value properties of P (n)SL(n) and p(n)SL(n), and give twosharper asymptotic formulas for them.Keywords F.Smarandache LCM function, mean value, asymptotic formula.§1. Introduction and ResultsFor any positive integer n, the famous F.Smarandache LCM function SL(n) defined as thesmallest positive integer k such that n | [1, 2, · · · , k], where [1, 2, · · · , k] denotes the leastcommon multiple of 1, 2, · · · , k. For example, the first few values of SL(n) are SL(1) = 1,SL(2) = 2, SL(3) = 3, SL(4) = 4, SL(5) = 5, SL(6) = 3, SL(7) = 7, SL(8) = 8, SL(9) = 9,SL(10) = 5, SL(11) = 11, SL(12) = 4, SL(13) = 13, SL(14) = 7, SL(15) = 5, · · · . About theelementary properties of SL(n), some authors had studied it, and obtained some interestingresults, see reference [2] and [3].For example, Lv Zhongtian [4] studied the mean value properties of SL(n), and provedthat for any fixed positive integer k and any real number x > 1, we have the asymptotic formula∑n≤xSL(n) =pi212· x2lnx+k∑i=2ci · x2lni x+O(x2lnk+1 x),where ci (i = 2, 3, · · · , k) are computable constants.Jianbin Chen [5] studied the value distribution properties of SL(n), and proved that forany real number x > 1, we have the asymptotic formula∑n≤x(SL(n)− P (n))2 = 25· ζ(52)· x52lnx+O(x52ln2 x),where ζ(s) is the Riemann zeta-function, and P (n) denotes the largest prime divisor of n.The main purpose of this paper is using the elementary methods to study the mean valueproperties of P (n)SL(n) and p(n)SL(n), and give two sharper asymptotic formulas for them.That is, we shall prove the following two conclusions:Vol. 3 On the mean value of the Smarandache LCM function 59Theorem 1. For any real number x > 1, we have the asymptotic formula∑n≤xP (n)SL(n) = x3 ·k∑i=1cilni x+O(x3lnk+1 x),where P (n) denotes the largest prime divisor of n, and ci (i = 1, 2, · · · , k) are computableconstants.Theorem 2. For any real number x > 1, we have the asymptotic formula∑n≤xp(n)SL(n) = x3 ·k∑i=1bilni x+O(x3lnk+1 x),where p(n) denotes the smallest prime divisor of n, bi (i = 1, 2, · · · , k) are computable constantsand b1 =13.Whether there exist an asymptotic formula for∑n≤xp(n)SL(n)and∑n≤xP (n)SL(n)is an openproblem.§2. Proof of the theoremsIn this section, we shall use the elementary methods to complete the proof the theorems.First we prove Theorem 1. In fact for any positive integer n > 1, let n = pα11 pα22 · · · pαss bethe factorization of n into prime powers, then from [2] we know thatSL(n) = max{pα11 , pα22 , · · · , pαss }. (1)Now we consider the summation ∑n≤xP (n)SL(n). (2)We separate all integer n in the interval [1, x] into four subsets A, B, C and D as follows:A: P (n) ≥ √n and n = m · P (n), m < P (n);B: n13 < P (n) ≤ √n and n = m · P 2(n), m < n 13 ;C: n13 < p1 < P (n) ≤√n and n = m · p1 · P (n), where p1 is a prime;D: P (n) ≤ n 13 .It is clear that if n ∈ A, then from (1) we know that SL(n) = P (n). Therefore, by theAbel’s summation formula (See Theorem 4.2 of [6]) and the Prime Theorem (See Theorem 3.2of [7]):pi(x) =k∑i=1ai · xlni x+O(xlnk+1 x),where ai (i = 1, 2, . . . , k) are computable constants and a1 = 1.60 Xiaoyan Li No. 3We have∑n∈AP (n)SL(n) =∑n∈AP (n)2 =∑m·p≤xm<pp2 =∑m≤√x∑m<p< xmp2=∑m≤√x[x2m2pi( xm)−∫ xmm2ypi(y)dy +O(m3)]=∑m≤√x(x3m3k∑i=1bilni xm+O(x3m3 · lnk+1 xm))= ζ(3) · x3 ·k∑i=1bilni x+O(x3lnk+1 x), (3)where ζ(s) is the Riemann zeta-function, b1 =13, bi (i = 2, 3, · · · , k) are computable constants.Similarly, if n ∈ C, then we also have SL(n) = P (n). So∑n∈CP (n)SL(n) =∑mp1p≤xm<p1<pp2 =∑m≤x 13∑m<p1≤√xm∑p1<p≤ xp1mp2=∑m≤x 13∑m<p1≤√xm[x2p21m2pi(xp1m)− p21pi(p1)−∫ xp1mp12ypi(y)dy]=∑m≤x 13x3m3k∑i=1di(m)lni xm+O(x3m3 lnk+1 xm)= x3 ·k∑i=1hilni x+O(x3lnk+1 x), (4)where hi (i = 1, 2, · · · , k) are computable constants.Now we estimate the error terms in set B. Using the same method of proving (3),we have∑n∈BP (n)SL(n) =∑m·p2≤xp3 =∑m≤x 13∑m<p<√xmp3= ζ(2) · x2 ·k∑i=1eilni x+O(x2lnk+1 x)= O(x2). (5)Finally, we estimate the error terms in set D. For any integer n ∈ D, let SL(n) = pα. WeVol. 3 On the mean value of the Smarandache LCM function 61assume that α ≥ 1. This time note that P (n) ≤ n 13 , we have∑n∈DP (n)SL(n) =∑mpα≤xα≥1,p≤x 13pα+1 ¿∑mpα≤xα≥1,p≤x 13p2α¿∑pα≤xα≥1,p≤x 13p2α∑m≤ xpα1¿ x∑pα≤xα≥1,p≤x 13pα ¿ x 73 . (6)Combining (2), (3), (4), (5) and (6) we may immediately obtain the asymptotic formula∑n≤xP (n)SL(n) =∑n∈AP (n)SL(n) +∑n∈BP (n)SL(n)+∑n∈CP (n)SL(n) +∑n∈DP (n)SL(n)= x3 ·k∑i=1cilni x+O(x3lnk+1 x).where P (n) denotes the largest prime divisor of n, and ci (i = 1, 2, · · · , k) are computableconstants.This proves Theorem 1.Now we prove Theorem 2. We separate all integer n in the interval [1, x] into four subsetsA¯, B¯, C¯ and D¯ as follows: A¯: n = 1; B¯: n = pα, α ≥ 1; C¯: n = pα11 pα22 , αi ≥ 1, (i = 1, 2); D¯:n = pα11 pα22 · · · pαss , αi ≥ 1, (i = 1, 2, · · · , s), s ≥ 3. p(n) denotes the smallest prime divisor ofn, p(1) = 0 and SL(1) = 1. Then we have∑n≤xp(n)SL(n) =∑n∈B¯p(n)SL(n) +∑n∈C¯p(n)SL(n) +∑n∈D¯p(n)SL(n). (7)Obviously if n ∈ B¯, then from (1) we know that SL(n) = pα. Therefore,∑n∈B¯p(n)SL(n) =∑pα≤xppα =∑p≤xp2 +∑pα≤xα≥2pα+1= x2pi(x)−∫ x322ypi(y)dy +O ∑2≤α≤ln x∑p≤x 1αp2α= x2pi(x)−∫ x322ypi(y)dy +O(x52)= x3 ·k∑i=1bilni x+O(x3lnk+1 x), (8)where b1 =13, bi (i = 2, 3, · · · , k) are computable constants.62 Xiaoyan Li No. 3If n ∈ C¯, then n = pα11 pα22 , where p1 < p2, and SL(n) ≥√n, so we have∑n∈C¯p(n)SL(n) =∑pα11 pα22 ≤xSL(pα11 pα22 )p1=∑pα11 ≤√x∑pα22 ≤ xpα11pα22 p1 +∑pα22 ≤√x∑pα11 ≤ xpα22pα11 p1 ¿ x114 . (9)Finally, we estimate the error terms in set D¯, this time, n = pα11 pα22 · · · pαss , where s ≥ 3.Therefore, n13 ≤ SL(n) ≤ √n, and p(n) ≤ n 13 , so we have∑n∈D¯p(n)SL(n)¿∑n≤xn13n12 ¿ x 116 . (10)Combining (1), (7), (8), (9) and (10) we may immediately obtain the asymptotic formula∑n≤xp(n)SL(n) = x3 ·k∑i=1bilni x+O(x3lnk+1 x),where p(n) denotes the smallest prime divisor of n, and b1 =13, bi (i = 2, 3, · · · , k) are com-putable constants.This completes the proof of Theorem 2.References[1] F.Smarandache, Only Problems, not solutions, Chicago, Xiquan Publ. House, 1993.[2] A.Murthy, Some notions on least common multiples, Smarandache Notions Journal,12(2001), 307-309.[3] Le Maohua, An equation concerning the Smarandache LCM function, SmarandacheNotions Journal, 14(2004), 186-188.[4] Lv Zhongtian, On the F.Smarandache LCM function and its mean value, ScientiaMagna, 3(2007), No. 1, 22-25.[5] Jianbin Chen, Value distribution of the F.Smarandache LCM function, Scientia Magna,3(2007), No. 2, 15-18.[6] Tom M. Apostol, Introduction to Analytic Number Theory, New York, Springer-Verlag,1976.[7] Pan Chengdong and Pan Chengbiao, The elementary proof of the prime theorem, Shang-hai Science and Technology Press, Shanghai, 1988.

On the mean value of the Smarandache LCM function

Abstract

Similar works

Full text

Available Versions

ZENODO