A NUMBER THEORETIC FUNCTION AND ITS MEAN VALUE ∗ Ren Ganglian Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China rgl70718@sina.com
Abstract
Let q ≥ 3 be a fixed positive integer, eq (n) denotes the largest exponent of power q which divides n. In this paper, we use the elementary method to study the properties of theX sequence eq (n), and give some sharper asymptotic formulas for the mean value ekq (n).
Keywords:
Largest exponent; Asymptotic formula; Mean value.
n≤x
§1. Introduction Let q ≥ 3 be a fixed positive integer, eq (n) denotes the largest exponent of power q which divides n. It is obvious that eq (n) = m, if q m |n, and q m+1 † n. In problem 68 of [3], Professor F.Smarandach asked us to study the properties of the sequence eq (n). About this problem, lv chuan in [2] had given the following result: If p is a prime, m ≥ 0 is an integer X n≤x
em p (n) =
³ ´ p−1 ap (m)x + O logm+1 x , p
where ap (m) is a computable number. The author had used the analytic method to consider the special case: p1 and p2 are two fixed distinct primes. That is, for any real number x ≥ 1, we have the asymptotic formula ³ ´ X x ep1 p2 (n) = + O x1/2+ε , (1) p1 p2 − 1 n≤x where ε is any fixed positive number. In this paper, we use the elementary method to improve the error Xterm of (1), and give some sharper asymptotic formula for the mean value ekq (n). That is we shall prove the following: ∗ This
work is supported by the N.S.F.(10271093) and the P.S.F. of P.R.China.
n≤x
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SCIENTIA MAGNA VOL.1, NO.1
Theorem 1. Let q ≥ 3 be any fixed positive integer, then for any real number x ≥ 1, we have the asymptotic formula X
x + O (log x) . q−1
eq (n) =
n≤x
. Theorem 2. If q ≥ 3 is any fixed positive integer, k ≥ 2 is an integer, then we have the asymptotic formula X
ekq (n) =
n≤x
³ ´ q−1 Bq (k)x + O logk+1 x , q
where Bq (k) is given by the recursion formulas: Bq (0) = Bq (k) =
1 q−1 ,
´ 1 ³k (1 )Bq (k − 1) + (k2 )Bq (k − 2) + · · · + (kk−1 )Bq (1) + Bq (0) + 1 . q−1
Taking q = p1 p2 in Theorem 1, where p1 , p2 are two fixed distinct primes, we may immediately obtain the following Corollary. For any real number x ≥ 1, we have the asymptotic formula X
ep1 p2 (n) =
n≤x
x p1 p2 − 1
+ O (log x) .
§2. Proof of the theorems In this section, we shall complete the Theorems. Let M = [x], the greatest integer ≤ x, S denotes the set of {1, 2, 3, · · · , M }. We distribute the integers of S into disjoint sets as follows. For each integer m ≥ 0, let A(m) = {n| eq (n) = m, 1 ≤ n ≤ M }. That is, A(m) contains those elements of S which satisfies: q m |n, but † n. Therefore if f (m) denotes the number of integers in A(m), we have
q m+1
¸
·
·
M M f (m) = m − m+1 q q
¸
So we have X
=
n≤x ∞ X m=1
eq (n) = µ·
m
X n≤M
¸
eq (n) = ·
M M − m+1 m q q
∞ X
mf (m)
m=0
¸¶
=
¸ ∞ · X M
m=1
qm
A number theoretic function and its mean value 1 ∞ X M
=
m=1
qm
Ã
∞ X x
=
m=1
+O
qm
X
∞ X 1
m=1
qm
1 + O
M m≤ log log q
+O
165 X
M m> log log q
!
+ O(
M qm
log M ) log q
x + O (log x) . q−1
=
This completes the proof of Theorem1. Before proving Theorem 2, we consider the series Bq (k) =
m=1
easy to show that Bq (0) = Bq (k) =
∞ X mk
∞ X 1
=
qm m=1
qm
, it is
1 , and Bq (k) satisfies the recursion formula q−1
´ 1 ³k (1 )Bq (k − 1) + (k2 )Bq (k − 2) + · · · + (kk−1 )Bq (1) + Bq (0) + 1 . q−1
Now we complete the proof of theorem2, with the same method as above, we have X
=
n≤x ∞ X
ekq (n) = µ·
mk
m=1
=
∞ X
µ
mk
m=1
X n≤M
¸
ekq (n) =
mk f (m)
m=0
·
M M − m+1 m q q
M M − m+1 m q q
∞ X
¶
¸¶
+O
X
M m≤ log log q
mk + O
X M m> log log q
M mk qm
=
∞ ∞ M ( log M + u)k X mk 1 (q − 1)M X log q k+1 + O(log M ) + O M m [ log ] q q qu q log q u=1 m=1
=
(q − 1)M Bq (k) + O(logk+1 M ) q õ
+O = =
log M log q
¶k X ∞ 1
µ
log M + (k1 ) u q log q u=1
(q − 1)M Bq (k) + O(logk+1 M ) q ³ ´ q−1 Bq (k)x + O logk+1 x . q
This completes the proof of Theorem2.
¶k−1 X ∞
∞ X u uk k + · · · + ( ) k qu qu u=1 u=1
!
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SCIENTIA MAGNA VOL.1, NO.1
References [1] Tom M.Apostol, Introduction to Analytic Number Theory, New York, Springer-Verlag, 1976. [2]Zhang Wenpeng, Research on Smarandache problems in number theory, Hexis, 2004. [3F.Smarandache, Only Problems, Not Solutions, Chicago, Xiquan Publ. House, 1993. [4] Pan Chengdong and Pan Chengbiao, Foundation of Analytic Number Theory, Beijing, Science Press, 1997.