The main purpose of this paper is to study the asymptotic property of the divisor product sequences, and obtain two interesting asymptotic formulas

ZHU, WEIYI

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ON THE DIVISOR PRODUCT SEQUENCES ZHU WEIYI College of Mathematics, Physics and Information Science, Zhejiang Normal University Jinhua, Zhejiang, P.R.China ABSTRACT. The main purpose of this paper is to study the asymptotic property of the divisor product sequences, and obtain two interesting asymptotic formulas. 1. INTRODUCTION AND RESULTS A natural number a is called a divisor product of n if it is the product of all positive divisors of n. vVe write it as Pd(n), it is easily to prove that Pd(n) = n where d(n) is the divisor function. We can also define the proper divisor product of n as the product of all positive divisors of n but n, we denote it by Pd(n), and () d.( 1'1.) 1 ( ) Pd n = n-:z-- , It is clear that the Pd n sequences is 1,2,3,8,5,36,7,64,27,100,11,1728,13,196,225"" ; The Pd( n) sequences is 1,1,1,2,1,6, 1,8,3, 10,1,144,1,14,15,64,1,324,1,1,400,21,···. In reference [1], Professor F, Smarandache asked us to study the properties of these two sequences. About these problems, it seems that none had studied them before. In this paper, we use the analytic methods to study the asymptotic properties of these sequences, and obtain two interesting asymptotic formulas. That is, we shall prove the following two Theorems. Theorem 1. For' any real number x ;::: 1, we have the asymptotic formula 1 1 2:p-( ) =lnlnx+C1+O(-ln ) < d n x n_x where C1 is a constant. Theorem 2. For any real number x ;::: 1, we have the asymptotic formula , 1 Inlnx 2: 7r(x) + (lnlnx)2 + B lnlnx + C2 + 0(-1-)' Pd(n) nx n:S;x· where 7r( x) is the number of all primes ::s; x , Band C2 are constants. Key words and phmses. Divisor products of ni Proper divisor products of n i Asymptotic formula .. 144 2. SOME LEMMAS To complete the proof of the theorems, we need following several lemmas. Lemma 1. For any real number x ~ 2, there is a constant A such that 1 1 2:-=lnlnx+A+O(-). < p lnx P x Proof. See Theorem 4.12 of reference [2]. Lemma 2. Let x ~ 2, then we have "" lnp 1 6- =lnx+C+O(-). Psx p lnx where C is constant. Proof. See reference [4]. Lemma 3. Let x ;::: 4, p and q are primes. Then we have the asymptotic formula "" 1 2 mInx L = (Inmx) +AInlnx + C3 + 0(--), pq~x pq mx where A and C3 are constants. Proof. From Lemma' 1 and Lemma 2 we have ( 2 1 1 1 1 2: =22:-2:-- 2:-pg P '" g P pq$x p$v'X q$. P$v'X) = 2 2: ~ (InInX + InCl ln p ) + A + 0(_1_)) mx lnx _ (Inlnx + A -In 2 + 0(_1_)) 2 lux ~ 2 '" ~ (In In x _ (In p + 1 (In p ) 2 + ~ ( In p ) 3 + .. . 1 (In p y~ + . ,. ) ) 6 p In x 2 In x 3 In x n In x psfi' + 2A ~ ~ + O(lnInx) (lnlnx + A -ln2 + 0(_1_))2 6 p lnx Inx psfi 2 In In x = (lnInx) + 2Alnlnx + C3 + O( lnx This proves Lemma 3. 145 3. PROOF OF THE THEOREMS In this section, we shall complete the proof of the Theorems. First we prove Theorem 2. Note that the definition of Pd(n), we can separate n into four parts according to d(n) = 2,3,4 or d(n) 2: 5. 1 2 , 3, d(n) =, 4, 2: 5, if n = p, Pd(n) 1; if n p2, Pd ( n) = P; ifn=PiPj or n p3; Pd(n) den) others, Pd(n) = n~-1. Then by Lemma 1) 2 and 3 we have 1 1 1 1 "-="1+" -+"'-+"'-+ 6 Pd(n) L G PiP' ~ P ~ p3 n::Sx p::Sx PiPj::SX J p 2 ::Sx pJ $x n::Sx,d(n)~5 1 -1 Inlnx = 7r(X) + (lnlnx)2 + 2Alnln~c + C3 + 0(--) + lnlnx + A~ , lnx In2+0(~11 )+C4+0(~)+C5+0( ~) nx X3 yX 2 In In x = 7r(x) + (lnlnx) + B lnlnx + C2 + O( X This completes the proof of Theorem 2. Similarly, we can also prove Theorem L Note that the definition of Pd(n), we have 1 This completes the proof of Theorem 1. REFERENCES 1. F. Smarndache, ONLY PROBLEMS, NOT SOLUTION!, Xiquan Publishing House, Chicago, 1993, pp. 24-25. 2. Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. 3. Pan Chengdong and Pan Chengbiao, Elementa1'1J Number Theory, Bejing University Press, Beijing', 1992, pp. 440-447. 4 .. LB.Rosser and L.Schoenfeld, Approximate formulas for some junctions of prime numbers, Illinois J .Math. 6 (1962), 64-94 .. 146 

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