ON THE MEAN VALUE OF THE SCBF FUNCTION

Transcription

1 ON THE MEAN VALUE OF THE SCBF FUNCTION Zhang Xiaobeng Deartment of Mathematics, Northwest University Xi an, Shaani, P.R.China Abstract Keywords: The main urose of this aer is using the elementary method to study the asymtotic roerties of the SCBF function on simle numbers, and give an interesting asymtotic formula for it. SCBF function; Mean value; Asymtotic formula.. Introduction In reference [], the Smarandache Sum of Comosites Between Factors function SCBF n is defined as: The sum of comosite numbers between the smallest rime factor of n and the largest rime factor of n. For eamle, SCBF 40, since 7 4 and the sum of the comosites between and 7 is: In reference []: A number n is called simle number if the roduct of its roer divisors is less than or eual to n. Let A denotes set of all simle numbers. That is, A {,, 4, 5, 6, 7, 8, 9, 0,,, 4, 5, 7, 9,, }. According to reference [], Jason Earls has studied the arithmetical roerties of SCBF n and roved that SCBF n is not a multilicative function. For eamle, SCBF and SCBF 4 SCBF He also got that if i and j are ositive integers then SCBF i 5 j 4, SCBF i 7 j 0, etc. In this aer, we use the elementary method to study the mean value roerties of SCBF n on simle numbers, and give an interesting asymtotic formula for it. That is, we shall rove the following: Theorem. Let, A denotes the set of all simle numbers. Then we have the asymtotic formula where B n n A. Some Lemmas SCBF n B ln + O is a constant, ln denotes the summation over all rimes. To comlete the roof of the theorem, we need the following lemmas:,

2 08 SCIENTIA MAGNA VOL., NO. Lemma. For any rime and ositive integer k, we have the asymtotic formula SCBF k 0. Proof. See reference []. Lemma. Let n A, then we have n, or n, or n, or n four case, where, denote the distinct rimes. Proof. First let n be a ositive integer, d n is the roduct of all ositive divisors of n, that is, d n d n d. dn is the roduct of all ositive divisors of n but n. That is, d n d n,d<nd. Then from the definition of d n we know that d n d n d. d n d n So from this formula we have dn d n d d n n d n n dn. d n where dn d n. Then we may immediately get dn n dn and d n d n,d<n d d n d n dn. n By the definition of the simle numbers, we get n dn n. Therefore, we have dn 4. This ineuality holds only for n, or n, or n, or n four cases. This comletes the roof of Lemma. Lemma. For any distinct rime and, we have the asymtotic formula SCBF + O ln ln Proof. From the definition of SCBF n, we have SCBF n, <n< < < ln where is a rime. Using the Abel s Identity [] and note that the asymtotic formula n n α α+ α + + Oα.

3 On the mean value of the SCBF function 09 we can get SCBF <n< n <n n n n n n + < < < < πtdt ln + ln + O + O π + π ln This comletes the roof of Lemma. Lemma 4. For real number, we have the asymtotic formula SCBF B ln + O ln, where and are two distinct rimes, B denotes the summation over all rimes.. is a constant, and Proof. From the definition of SCBF n and Lemma, Lemma, we get SCBF SCBF SCBF,< SCBF < < Noting that π ln + O < ln ln + ln + O, using Abel s Identity [] we get π π πttdt ln ln + O ln ln.

4 0 SCIENTIA MAGNA VOL., NO. and < ln A f Af ln Atft dt ln 9 ln + O ln, where A <, f ln ln ln + C + O ln. From reference [], we know that, where C is a comutable constant. And then we also get ln + O ln and ln + O ln. Using the same method, we obtain ln ln + O ln and ln ln + O ln. Noting that + ln ln ln + ln ln + + lnm ln m +, then we get the ln following two formulae: < ln ln C ln + O ln + O ln + O + ln ln + ln ln ln ; ln ln + + ln ln + ln ln +

5 On the mean value of the SCBF function < ln 9 ln ln C ln + O where C C So we have where B ln 9 ln + O + ln ln + ln ln + O ln,. < < < + ln B ln + O. Proof of the theorem SCBF ln ln ln + ln + ln + O < + O < ln. This roves Lemma 4., ln < ln ln < ln In this section, we comlete the roof of Theorem. According to the definition of simle numbers and Lemma, we have SCBF n n n A

6 SCIENTIA MAGNA VOL., NO. SCBF + SCBF + SCBF + SCBF. And then, using Lemma and Lemma 4 we obtain SCBF n SCBF n n A This comletes the roof of Theorem. B ln + O ln. Acknowledgments The author eress his gratitude to his suervisor Professor Zhang Weneng for his very helful and detailed instructions. References [] Jason Earls, The Smarandache Sum of Comosites Between Factors Function, Smarandache Notions Journal, 4 004, [] F. Smarandache, Only Problems, Not Solutions, Chicago, Xiuan Publ. House, 99. [] Tom M. Astol, Introduction to Analytic Number Theory, New York, Sringer-Verlag, 976.