sdf-ln41
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On the other hand, a number n is called simple number if theproduct of its proper divisors is less than or equal to n . Generallyspeaking, n has the form: n = p, or p2 , or p3 , or pq, where p andq are distinct primes. Professor F. Smarandache [1] also asked us tostudy this sequence. Let A be the set of all the simple number. In thispaper, we study the mean value of the Smarandache double factorialfunction, and give a sharp asymptotic formula. That is, we shall provethe following:
Theorem If x ≥ 2, then for any positive integer k we have
n An ≤x
df (n ) =x 2
log x52 + p
1 p2 +
k−1
m =1
cm
logm x + Ox 2
logk +1 x ,
where p
denotes the summation over all prime numbers, and cm (m =
1, 2, · · · , k − 1) are computable constants.
§2. Proof of the Theorem
In this section, we complete the proof of the Theorem.
It is obvious that df ( p) = p, df (2 p) = 2 p, df ( pq) = max { p,q}, if p = q and p, q = 2. Also we have df ( pα ) ≤ (2α − 1) p. Then we have
n An ≤x
df (n ) = p≤x
df ( p) + p2 ≤x
df ( p2 ) + p3 ≤x
df ( p3 ) + pq≤x p= q
df ( pq)
= p≤x
p + 3 p≤x
12
p + 5 p≤x
13
p + pq≤x p= q
df ( pq). (1)
Leta (n ) = 1, if n is prime;
0, otherwise ,then for any positive integer k we have
n ≤y
a (n ) = π (y) =y
log y1 +
k−1
m =1
m !logm y
+ Oy
logk +1 y. (2)
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By Abel’s identity we have
p≤y
p =n ≤y
a (n )n = π (y)y − y
2π (t )dt
=y2
log y1 +
k−1
m =1
m !logm y
+ O (y2
logk +1 y)
− y
2
tlog t
+k−1
m =1
m !tlogm +1 t
+ Ot
logk +1 tdt
=y2
log y12
+k−1
m =1
a m
logm y+ O
y2
logk +1 y. (3)
Then from (1) and (3) we have
n An ≤x
df (n ) =x 2
log x12
+k−1
m =1
a m
logm x+
pq≤x p= q
df ( pq) + Ox 2
logk +1 x. (4)
On the other hand, by (3) we have
pq≤x p= q
df ( pq) = 22<q ≤x
df (2q) + pq≤x p= q
p, q=2
df ( pq)
= 42<q ≤x
q + 22<p ≤√x
p2<q<p
+22<p ≤√x p<q ≤x
p
q −2<p ≤√x 2<q ≤√x
df ( pq)
=x 2
log x2 +
k−1
m =1
4a m
logm x+ 2
2<p ≤√x p<q ≤x
p
q + Ox 2
logk +1 x.
That is to say,
n An ≤x
df (n ) =x 2
log x52 +
k−1
m =1
5a m
logm x + 22<p ≤√x p<q ≤x
p
q + Ox 2
logk +1 x . (5)
From (3) we also have
22<p ≤√x p<q ≤x
p
q = 22<p ≤√x
x 2
p2 log x p
12
+k−1
m =1
a m
logm x p
+ Ox 2
p2 logk +1 x p
−p2
log p12
+k−1
m =1
a m
logm x p
+ Op2
logk +1 p.
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Noting that
1log x
p=
1
log x 1 − log plog x
=1
log x1 +
log plog x
+log2 plog2 x
+ · · · ,
then we have
22<p ≤√x p<q ≤x
p
q = 22<p ≤√x
x 2
p2 log x1 +
∞
l=1
logl p
logl x
×12
+k−1
m =1
a m
logm x1 +
∞
l=1
logl p
logl x
m
+ Ox 2
p2 log x+ O
p2
log p
=x 2
log x p
1 p2 +
k−1
m =1
bm
logm x+ O
x 2
logk +1 x, (6)
so from (5) and (6) we get
n An ≤x
df (n ) = p≤x
df ( p) + p2 ≤x
df ( p2 ) + p3 ≤x
df ( p3 ) + pq≤x p= q
df ( pq)
= p
≤x
p + 3 p
≤x
12
p + 5 p
≤x
13
p + pq
≤x
p= q
df ( pq). (7)
This completes the proof of the Theorem.
References
[1] F.Smarandache. Only problems, not Solutions. Chicago: Xi-quan Publ. House, 1993.
[2] C.Dumitrescu and V.Seleacu. Some notions and questions inNumber Theory. Glendale: Erhus Univ. Press, 1994.
[3] Felice Russo. A set of new Smarandache functions, sequencesand conjectures in number theory. American Research Press, 2000.
[4] Felice Russo. Five properties of the Smarandache Double Fac-torial Function. Smarandache Notions Journal, 2002, 13: 183-185.
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