On a note of the Smarandache power function 1

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1 Scientia Magna Vol , No. 3, On a note of the Smarandache ower function Wei Huang and Jiaolian Zhao Deartment of Basis, Baoji Vocational and Technical College, Baoji 7203, China Deartment of Mathematics, Weinan Teacher s University, Weinan 74000, China whuangwei@63.com Abstract For any ositive integer n, the Smarandache ower function SP n is defined as the smallest ositive integer m such that n m m, where m and n have the same rime divisors. The main urose of this aer is to study the distribution roerties of the k th ower of SP n by analytic methods, obtain three asymtotic formulas of SP n k, ϕsp n k and dsp n k k >, and sulement the relate conclusions in some references. Keywords Smarandache ower function, the k th ower, mean value, asymtotic formula.. Introduction and results For any ositive integer n, we define the Smarandache ower function SP n as the smallest ositive integer m such that n m m, where n and m have the same rime divisors. That is, { SP n min m : n m m, m N +, }. m n If n runs through all natural numbers, then we can get the Smarandache ower function sequence SP n:, 2, 3, 2, 5, 6, 7, 4, 3, 0,, 6, 3, 4, 5, 4, 7, 6, 9, 0,, Let n α α2 2 α k k, denotes the factorization of n into rime owers. If α i < i, for all α i i, 2,, r, then we have SP n Un, where Un, denotes the roduct over all different rime n n divisors of n. It is clear that SP n is not a multilicative function. In reference [], Professor F. Smarandache asked us to study the roerties of the sequence SP n. He has done the reliminary research about this question literature [2] [4], has obtained some imortant conclusions. And literature [2] has studied an average value, obtained the asymtotic formula: SP n 2 x2 + O x 32 +ε. + This aer was artially suorted by National Natural Science Foundation of China 06755, The Natural Sciences Funding of Project Shaanxi Province SJ08A28 and The Educational Sciences Funding Project of Shaanxi Province 200JK54.

2 94 Wei Huang and Jiaolian Zhao No. 3 Literature [3] has studied the infinite sequence astringency, has given the identical equation: 2 s + 2 s, k, 2; ζs µn SP n k s 2 s + 2 s ζs 2s 4 s, k 3; n 2 s + 2 s ζs 2s 4 s + 3s 9 s, k 4, 5. And literature [4] has studied the equation SP n k φn, k, 2, 3 solubility φn is the Euler function, and has given all ositive integer solution. Namely the equation SP n φn only has 4 ositive integer solutions n, 4, 8, 8; Equation SP n 3 φn to have and only has 3 ositive integer solutions n, 6, 8. In this aer, we shall use the analysis method to study the distribution roerties of the k th ower of SP n, gave SP n k, ϕsp n k and dsp n k k >, some interesting asymtotic formula, has romoted the literature [2] conclusion. Secifically as follows: Theorem.. For any random real number x 3 and given real number k k > 0, we have the asymtotic formula: SP n k ζk + k + ζ2 xk+ k + Ox k+ 2 +ε ; + SP n k n ζk + kζ2 xk k + Ox k+ 2 +ε, + where ζk is the Riemann zeta-function, ε denotes any fixed ositive number, and denotes the roduct over all rimes. Corollary.. For any random real number x 3 and given real number k > 0 we have the asymtotic formula: SP n k 6k ζ +k k k + π 2 x +k k + O x k k +ε. k Secifically, we have 4ζ 3 SP n 2 2 π 2 x ζ 4 SP n 3 3 2π 2 x n SP 2 x2 SP n 2 6ζ3 3π 2 + Ox +ε ; + Ox 5 6 +ε. Corollary.2. For any random real number x 3, and k, 2, 3. We have the asymtotic formula: + Ox 3 2 +ε ; + x Ox 5 2 +ε ;

3 Vol. 6 On a note of the Smarandache ower function 95 SP n 3 π2 60 x4 ϕ SP n k ζk + k + ζ2 xk+ 3 + Ox 7 2 +ε. + Theorem.2. For any random real number x 3, we have the asymtotic formula: + Ox k+ 2 +ε, + k where ϕn is the Euler function Theorem.3. For any random real number x 3, we have the asymtotic formula: d SP n k B 0 x ln k x + B x ln k x + B 2 x ln k 2 x + + B k x ln x + B k x + Ox 2 +ε. where dn is the Dirichlet divisor function and B 0, B, B 2,, B k, B k is comutable constant. 2. Lemmas and roofs Suose s σ + it and let n α α2 2 α k k, Un n. Before the roofs of the theorem, the following Lemmas will be useful. Lemma 2.. For any random real number x 3 and given real number k, we have the asymtotic formula: Un k ζk + k + ζ2 xk+ Proof. Let Dirichlet s series As n Un k n s, + k + Ox k+ 2 +ε. for any real number s >, it is clear that As is absolutely convergent. Because Un is the multilicative function, if σ > k +, so from the Euler s roduct formula [5] we have As Un k n n s U m k m0 ms + k s + k 2s + ζsζs k ζ2s 2k k + s k, where ζs is the Riemann zeta-function. Letting Rk, Un n, n Un k n < ζσ k. σ. If σ > k + k + s k

4 96 Wei Huang and Jiaolian Zhao No. 3 Therefore by Perron s formula [5] with an Un k, s 0 0, b k + 3 2, T xk+ 2, Hx x, Bσ ζσ k, then we have Un k 2πi where hk. k + To estimate the main term 2πi k it k+ 2 it ζsζs k ζ2s 2k hsxs s ds + Oxk+ 2 +ε, k it k+ 2 it ζsζs k ζ2s 2k hsxs s ds, we move the integral line from s k ± it to k + 2 ± it, then the function ζsζs k ζ2s 2k hsxs s have a first-order ole oint at s k + with residue ζsζs k Lx Res sk+ ζ2s 2k hs ζsζs k lim s k s k+ ζ2s 2k hsxs s ζk + k + ζs xk+ hk. Taking T x k+ 2, we can easily get the estimate k it k it ζsζs k 2πi ζ2s 2k hsxs s ds x2k+ x k+ 2, T k+ 2 +it + 2πi k+ 2 it k+ 2 +it ζsζs k k+ 2 it ζ2s 2k hsxs s ds xk+ 2 +ε. We may immediately obtain the asymtotic formula Un k ζk + k + ζ2 xk+ + k + Ox k+ 2 +ε, this comletes the roof of the Lemma 2.. Lemma 2.2. For any random real number x 3 and given real number k, and ositive integer α, then we have α k ln 2k+2 x. α x α> Proof. Because α >, so < α x, then < ln x ln x < ln x, α ln ln,

5 Vol. 6 On a note of the Smarandache ower function 97 also, n k xk+ k + + Oxk. Thus, α k α x α> Considering πx Therefore ln x Thus k x ln x k α ln x ln α k ln k+ x ln x, by virtue of [5], πx x x ln x + O k πxx k k x lnk x ln x k + + Olnk x k 2 α k α x α> ln x k α ln x ln α k ln k+ x This comletes the roof of the Lemma 2.2. x 2 k ln k+ lnk+ x ln 2 x πtt k dt. t k ln x ln t dt + O 2 ln x ln x k.. we can get from the Able t k ln 2 t dt lnk x k + + Olnk x. k ln k+ lnk+ x ln x k ln 2k+2 x. 3. Proof of the theorem [ SP n k Un k] In this section, we shall comlete the roof of the theorem. Proof of Theorem.. Let A { n n α α2 2 α k k, α i i, i, 2,, r }. When n A : SP n Un; When n N + : SP n Un, thus n SP k Un k SP n>un SP n k. By the [2] known, there is integer α and rime numbers, so SP n < α, then we can get according to Lemma 2.2 SP n k < α k x ln 2k+2 x. Therefore SP n>un From the Lemma 2. we have SP n k SP n>un SP n k ζk + k + ζ2 xk+ α <x α> Un k x ln 2k+2 x. k + + Oxk+ 2 +ε + Ox ln 2k+ x ζk + k + ζ2 xk+ k + + Oxk+ 2 +ε.

6 98 Wei Huang and Jiaolian Zhao No. 3 This roves Theorem.. Proof of Corollary. According to Theorem., taking k k the Corollary. can be obtained. Take k, 2, 3, and ζ2 π2 π4, ζ4, we can achieve Corollary.2. Obviously 6 90 so is theorem [2]. Using the similar method to comlete the roofs of Theorem.2 and Theorem.3. Acknowledgments The authors exress many thanks to the anonymous referees for valuable suggestions and comments. References [] F. Smarandache, Collected aers, Bucharest, Temus Publ. Hse., 998. [2] Zhefeng Xu, On the mean value of the Smarandache ower function, Acta Mathematics Sinica Chinese series, , No., [3] Huanqin Zhou, An infinite series involving the Smarandache ower function SP n, Scientia Magna, 22006, [4] Chengliang Tian and Xiaoyan Li, On the Smarandache ower function and Euler totient function, Scientia Magna, 42008, No., [5] Chengdong Pan and Chengbiao Pan, The Elementary Number Theory, Beijing University Press, Beijing, [6] Tom M. Aostol, Introduction to Analytic Number Theory, New York, Sringer-Verlag, 976.