Studying the asymptotic properties of a mean value, and giving an interesting asymptotic formula for it

HONGYAN, L.

WENPENG, ZHANG

English

ZENODO

A NUMBER THEORETIC FUNCTION AND ITS MEAN VALUE PROPERTY* Lru HONGYAN AND ZHANG WENPENG Department of Mathematics, Northwest University Xi'an, Shaanxi, P.R.China ABSTRACT. Let p be a prime, n be any positive integer, a(n,p) denotes the power of p in the factorization of n!. In this paper, we study the asymptotic properties of the mean value L a(n, p), and give an interesting asymptotic formula for it. p$n 1. INTRODUCTION Let p be a prime, ep ( n) denotes the largest exponent ( of power p ) which divides n, a(n,p) = L ep(k). In problem 68 of [1], Professor F.Smarandach asked us to k=:;n study the properties of the sequences ep(n). This problem is interesting because there are close relations between ep ( n) and the factorization of n!. In fact, 0'( n, p) is the power of p in the factorization of nL In this paper, we use the elementary methods to study the asymptotic properties of the mean value L 0'( n, p), and give p=:;n an interesting asymptotic formula for it. That is, we shall prove the following: Theorem. For any prime p and any fixed positive integer n, we have the asymp-totic formula '" a( n p) = n In In n + en 4- C1 ~ + C2 ~ + ... + Ck ~ + 0 ( n ). L< ' ' Inn ln2 n Inkn lnk+1n p_n ° where k is any fixed positive integer, Ci (i = 1,2,··· ,) are some comp~Liable con-stants. 2. PROOF OF THE THEOREM In this section, we complete the proof of the Theorem. First for any prime p and any fixed positive integer n, we let a(n,p) denote the sum of the base p digits of n. That is, if n 0= alpO'l + a2p0'2 + ... + aspO'· with as > as-I> ... > 0'1 2: 0, where s 1 ~ ai ~ p-1, i = 1,2,··· ,8, then a(n,p) = Lai, and for this number theoretic i=l function, we have the following two simple Lemmas: Key words and phraseso A new number theoretic function; Mean value; Asymptotic formula. * This work is support.ed by the N.S.F. and the P.S.F. of P.R.China. 155 Lemma 1. For any integer n 2: 1, we have the identity +00 [n] 1 O:'p(n) == O:'(n) == ~ ~ = -- (n - a(n,p) , ~ p' p-1 . :=1 where [x] denotes the greatest integer not exceeding x. Proof. From the properties of [x] we know that [;] = [a1PQ1 + azpa:t ... + aspo. ] _ { t ajpOlj -i, if O:'k-1 < i ~ O:'k - j=k 0, if i > O:'s. So from this formula we have +00 [] +00 [a Ol Ol ] = ~ ~ _ ~ alP 1 + a2P 2 + ... + asp • 0:'( n) - 6 i - 6 i ;=1 p i=1 P s OJ S = L L ajpcxj-k = L aj (1 + p + p2 + ... + pOlj-1) j=lk=l j=1 s Ol· lIs ~ pJ- ~= 6 aj' -1 = ~ ~ (ajpQj - aj) j=1 p P i=l 1 = -- (n - a(n,p». p-l This completes the proof of Lemma 1. Lemma 2. For any positive integer n, we have the estimate p-1 a(n,p):S:; lnp In n. Proof. Let n = a1pO l + a2pOl2 + ... + aspo. with as > as-I> ... > a1 2: 0, where 1 ~ ai :s:; p - 1, i = 1,2"" , s. Then from the definition of a(n,p) we have s s (1) a(n,p) = L ai :s:; L(p -1) = (p - l)s. i=l i=l On the other hand, using the mathematical induction we can easily get the inequal-ity or (2) In(n/a s ) Inn s< <-. - lnp - Inp 156 Combining (1) and (2) we immediately get the estimate . p-1 a(n,p) ~ -I-Inn. np This proves Lemma 2. Now we use Lemma I and Lemma 2 to complete the proof of the Theorem. First, we separate the summation in the Theorem into two parts. (3) L a(n,p) = L a(n,p) + L a(n,p). v'n<p$n For the first part, from Lemma 1 we have 1 L a(n,p) = L p-1 (n - a(n,p)) p$v'n p$v'n "" (1 1) "" a(n,p) =n 6 ;+ p(p-I) - 6 p-1 p$v'n p$v'n _ n ("" ~ + "" 1 + 0 ("" ~)) _ "" a(n,p) - 6 p 6 pep _ 1) 6 m 2 L.,; p - 1 p$vfn p m>v'n p$v'n (4) = n ( rfii .!.d?r(x) + A + 0 (.2..)) _ "" a(n,p). J~ x yin 6 p-1 2 p$v'n where ?rex) denotes the number of all prime not exceeding x. For ?rex), we have the asymptotic formula (5) and (6) a31 an a3k (1) = In In n + B + -1 - + :2 + ... + -k- + 0 k+l . n 11 In n In 17 In 17 157 From Lemma 2 we have (7) "a(n,p) "Inn ,,1 L ~ 1::; ~ -1 - = Inn ~ -1 -::; Inn 1::; y'nlnn. p - np np p~Vn p~Vn p~Vn p~Vn Combining (4), (6) and (7) we obtain " n n ~ a(n,p) = nlnlnn + con + a 31 -1 - + a32-?-c nn In- n P<v n (8) + ... + a3k + + 0 ( k:l ). In n In n For the second part, we have (9) = .L 7r (:) - [v'n]7r(vn). m~..;n Applying Euler's summation formula ( see [2] Theorem 3.1 ) and the expansion into power-series we have = f (r -1 + s) (L InS m ) O r - 1 em 1ns+r n s= m~vn 158 From this and (5) we get =n L ( __ 1 __ +a2 1 + ... m(In n - In m) m(In n - In m )2 m~y'n + ak+I +0 1 (1)) m(In n - In m )k+I m(ln n - In m )k+2 =n bo+-+--+···+--+O ( bi ~ bk (1)) In n In2 n Ink n Ink+I n (10) = bon+bI - +b2-- + ... +bk --+0 n n n (n) In n In2 n Ink n Ink+I n and (11) n n n (n) = a4II-- + a42-2- + ... + a4k-k- + 0 HI . nn Inn Inn In n Combining (9), (10) and (11) we have (12) '" a(n,p) = bon + aS1lnn + aS2+ + ... + aSk+ + 0 ( k:l )-. r.:~< n In n In n In n vn<p_n From (3), (8) and (12) we obtain the asymptotic formula '" n n n" (n) ~ a( n, p) = n In In n + cn + CI -In + c2 -2 - + ... + Ck -k - + 0 HI . < n Inn Inn In n p_n This completes the proof of the Theorem. REFERENCES 1. F. Smarandache, Only problems, not Solutions, Xiquan Pub!. House, Chicago, 1993, pp. 54-55. 2. Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. 3. R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1981. 4. "Smarandache Sequences" at http://www.gaIlup.unm.edursmarandache/snaqint.txt. 5. "Smarandache Sequences" at http://www.gaIlup.unm.edursmarandache/snaqint2.txt. 6. "Smarandache Sequences" at http://www.gaIlup.unm.edur smarandache/snaqint3. txt. 159 

Only problems, not Solutions, Xiquan Pub!. House,

A NUMBER THEORETIC FUNCTION AND ITS MEAN VALUE PROPERTY

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A NUMBER THEORETIC FUNCTION AND ITS MEAN VALUE PROPERTY

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