Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 4, 171 - 178
Asymptotic Formulas
Composite Numbers
Rafael Jakimczuk
División Matemática, Universidad Nacional de Luján
Buenos Aires, Argentina
jakimczu@mail.unlu.edu.ar
In memory of my sister Fedra Marina Jakimczuk (1970-2010)
Abstract
In this article we study the distribution of certain composite numbers
which have in their prime factorization a fixed number of different prime
factors, the exponents being fixed.
Let us consider the sequence An of composite numbers whose prime
factorization is of the form pa11 pa22 . . . pat t pht+1 . Where a1 ≥ a2 ≥ . . . ≥
at > h ≥ 1 are positive integers fixed and p1 , p2 , . . . , pt+1 are different
primes. Let A(x) be the number of these numbers not exceeding x. In
this article we prove that
1
hx h
,
A(x) ∼ α
log x
where α is defined in this article.
Mathematics Subject Classification: 11B99, 11N25
Keywords: Composite numbers, counting function, asymptotic formula
1
Introduction, notation and lemmas.
Let n be a number such that its prime factorization if of the form
n = pa11 pa22 . . . par r ,
where ai ≥ q ≥ 2 (i = 1, 2, . . . , r) and p1 , p2 , . . . , pr (r ≥ 1) are the different
primes in the factorization.
These number are well known, they are called powerful numbers or q-ful
numbers.
R. Jakimczuk
172
There exist various studies on the distribution of these numbers using not
elementary methods (see [1]).
Let Cn,q be the sequence of these numbers and let Cq (x) be the number of
these numbers that do not exceed x. It is well known ( see [2] for an elementary
proof) that
Cn,q ∼ cq nq ,
(1)
1
Cq (x) ∼ bq x q ,
(2)
where bq and cq are constants depending of q.
From (1) we can obtain without difficulty the following lemma.
Lemma 1.1 The following series are convergent (q ≥ 2)
∞
i=1
1
(Cn,q )
1
q−1
,
∞
log Cn,q
i=1
(Cn,q ) q−1
1
.
Let us consider the sequence En of the numbers whose prime factorization
is of the form
pa11 pa22 . . . pat t ,
where a1 ≥ a2 ≥ . . . ≥ at ≥ 2 (t ≥ 1) are positive integers fixed and the pj
(j = 1, 2, . . . , t) are different primes.
For example the sequence En of the numbers of the form p91 p52 p53 p34 where
p1 , p2 , p3 , p4 are different primes. In this case a1 = 9, a2 = 5, a3 = 5, a4 = 3, t =
4.
We shall denote these numbers in the compact form E.
The number of these numbers not exceeding x we shall denote E(x).
Let us consider the sequence An of the numbers whose prime factorization
is of the form
pa11 pa22 . . . pat t pht+1 ,
where a1 ≥ a2 ≥ . . . ≥ at > h ≥ 1 (t ≥ 1) are positive integers fixed and the
pj (j = 1, 2, . . . , t + 1) are different primes.
For example the sequence An of the numbers of the form p91 p52 p53 p34 p25 where
p1 , p2 , p3 , p4 , p5 are different primes. In this case a1 = 9, a2 = 5, a3 = 5, a4 =
3, h = 2, t = 4.
We shall denote these numbers in the compact form Eph where E denotes
the numbers of the form pa11 pa22 . . . pat t (see above) and ph denotes pht+1 .
The number of these numbers not exceeding x we shall denote A(x).
Since in this case the E numbers are (h+1)-ful numbers, lemma 1.1 imply
that the following series are convergent
∞
n=1
1
1
h
En
= AE,h ,
∞
log En
n=1
Enh
1
= BE,h .
(3)
Composite numbers
173
On the other hand (2) imply that from a certain value of x we have
1
E(x) ≤ (1 + )bh+1 x h+1
( > 0).
(4)
In theorem 2.1 we shall prove that
1
hx h
.
A(x) ∼ AE,h
log x
Let π(x) be the number of primes not exceeding x. We shall need the prime
number theorem which we shall use as a lemma.
Lemma 1.2 The following formula holds
x
x
π(x) =
+ f (x)
,
log x
log x
where |f (x)| ≤ M if x ≥ 2 and f (x) → 0.
2
Main results
Theorem 2.1 The following asymptotic formula holds
1
hx h
A(x) ∼ AE,h
.
log x
Proof. We have
(5)
Eph ≤ x,
x
ph ≤ ,
E
x
x
E ≤ h ≤ h,
p
2
1
xh
1
Eh
Therefore (lemma 1.2)
A(x) =
E≤
=
E≤
+
x
2h
E≤
=
x
2h
x
2h
x
ph ≤ E
⎛
≥ 2.
1 − F1 (x) =
E≤
1
x
2h
⎞
1 − F1 (x)
1
p≤ x h1
Eh
1
xh
xh
1
1 π ⎝ 1 ⎠ − F1 (x) =
1
h
Eh
E≤ xh E h log x 1
2
⎛
f⎝
1
h
⎞
Eh
1
h
x ⎠x
1
1 − F1 (x)
1
1
E h E h log x h1
Eh
1
h
1
hx 1
+ G1 (x) − F1 (x).
1
log x E≤ x E h 1 − log E h1
1
2h
log x h
(6)
R. Jakimczuk
174
Substituting x = 2h En into
E≤
we obtain the sequence
1
1
log E h
1
log x h
1
1
h
.
1
Ei 1 −
i=1
1
E 1−
x
2h
n
1
1
h
log Eih
(7)
1
log 2Enh
Note that if Ei ≤ En then
1
1
1
1
Eih 1 −
1
log Eih
≤
1
Eih 1 −
1
log 2Enh
1
1 log 2 + log Eih
1 log Ei
= 1 +
= 1
1
log 2
h log 2 E h
Eih
Eih
i
1
1
1
log Eih
1
log 2Eih
(8)
and if Ei is fixed then
lim
1
n→∞
1
1
h
Ei 1 −
1
log Eih
1
log 2Enh
=
1
1
Eih
.
(9)
We have (see (7))
n
i=1
1
1
1
h
Ei 1 −
1
log Eih
1
log 2Enh
=
k
i=1
1
1
1
h
Ei 1 −
1
log Eih
1
log 2Enh
n
+
i=k+1
1
1
h
Ei 1 −
1
1
log Eih
,
(10)
1
log 2Enh
where (see (8))
n
1
1
i=k+1
1
Eih 1 −
1
log Eih
n
1
i=k+1
Eih
≤
1
1
n
1
log Ei
.
h log 2 i=k+1 E h1
+
(11)
i
log 2Enh
There exists k such that ( > 0) (see (3))
AE,h − <
k
1
i=1
Eih
1
< AE,h ,
∞
log Ei
1
< .
h log 2 i=k+1 E h1
(12)
(13)
i
If n ≥ k + 1, (11), (12) and (13) give
0≤
n
i=k+1
1
1
h
Ei 1 −
1
1
log Eih
1
log 2Enh
≤ 2.
(14)
Composite numbers
175
On the other hand (see(9))
lim
k
n→∞
i=1
1
1
1
h
Ei 1 −
=
1
log Eih
1
log 2Enh
k
1
i=1
Eih
1
.
Consequently there exists n > k + 1 such that for all n ≥ n we have
k
i=1
1
1
h
Ei
−≤
k
1
1
1
h
1
log Eih
1
log 2Enh
Ei 1 −
i=1
≤
k
i=1
1
1
h
Ei
+ .
(15)
Equations (12) and (15) give
k
AE,h − 2 ≤
i=1
1
1
1
h
≤ AE,h + .
1
Ei 1 −
log Eih
(16)
1
log 2Enh
Therefore for all n ≥ n we have (see (10), (14) and (16))
AE,h − 3 ≤
Consequently
lim
n→∞
n
i=1
n
1
1
1
h
Ei 1 −
1
Ei 1 −
i=1
(17)
1
log 2Enh
1
1
h
≤ AE,h + 3.
1
log Eih
1
log Eih
= AE,h .
(18)
1
log 2Enh
Now , we have
⎛
⎜ n
⎜
⎜
lim ⎜
n→∞ ⎜
⎝i=1
⎞
1
1
Eih 1 −
⎛
=
⎜ n
⎜
⎜
lim ⎜
n→∞ ⎜
⎝i=1
1
1
h
Ei 1 −
1
1
log Eih
−
n+1
i=1
1
log 2Enh
1
1
log Eih
−
n
i=1
1
log 2Enh
1
1
1
Eih 1 −
1
log Eih
1
⎟
⎟
⎟
⎟
⎟
⎠
h
log 2En+1
⎞
1
1
1
1
h
Ei 1 −
1
log Eih
1
−
1
1
h
En+1
⎟
h
⎟
log 2En+1
⎟
⎟
log 2 ⎟
⎠
h
log 2En+1
= 0
(19)
and
1
lim
n→∞
1
1
h
En+1
h
log 2En+1
= 0.
log 2
(20)
R. Jakimczuk
176
Equations (19) and (20) give
⎛
⎞
⎜
n
⎜
⎜
lim ⎜
n→∞ ⎜
⎝ i=1
1
1
1
Eih 1 −
−
1
log Eih
n
1
1
Eih 1 −
i=1
1
log 2Enh
⎟
⎟
⎟
⎟
⎟
⎠
1
1
log Eih
1
= 0.
(21)
h
log 2En+1
Therefore (see (18))
lim
n
n→∞
lim
n→∞
1
1
Ei 1 −
i=1
n
i=1
1
1
h
log Eih
= AE,h ,
(22)
= AE,h .
(23)
1
log 2Enh
1
1
1
h
Ei 1 −
1
log Eih
1
h
log 2En+1
The function of x (Ei fixed, Ei ≤ En )
1
1
1
h
Ei 1 −
(24)
1
log Eih
1
log x h
is decreasing in the interval 2h En , 2h En+1 . Therefore if x ∈ 2h En , 2h En+1
we have
n
i=1
1
1
h
1
Ei 1 −
1
log Eih
1
h
log 2En+1
≤
n
i=1
1
1
1
h
Ei 1 −
1
log Eih
1
log x h
≤
n
i=1
1
1
1
h
Ei 1 −
1
log Eih
.
(25)
1
log 2Enh
Consequently (22), (23) and (25) give
lim
x→∞
E≤
x
2h
1
1
1
h
E 1−
1
log E h
= AE,h .
(26)
1
log x h
There exists x0 such that (lemma 1.2)
⎛
⎞
1
h
x
f ⎝
⎠
1
Eh ⎛
⎞
1
h
x
f ⎝
⎠
1
Eh 1
< if
xh
E
1
h
that is if
E≤
< x0 ,
that is if
x
x
< E ≤ h.
h
x0
2
1
≤M
if
2≤
xh
E
1
h
x
,
xh0
≥ x0 ,
Composite numbers
177
Therefore (see(6))
E≤ x
2h
|G1 (x)| =
⎛
f⎝
1
h
⎞
x ⎠x
1
1
Eh Eh
⎛
⎞
1
h
x
f ⎝
⎠
1
Eh E≤ x ≤
x
≤ E≤ xh
x
0
1
xh
E log
1
E log
1
1
h
Eh
1
h
1
h
2h
1
1 log x h1 1
h
1
xh
1
Eh
1
xh
1
Eh
+ Mx0
x
<E≤ xh
2
xh
0
1
1
hx h 1
=
1
1 + Mx0
log x E≤ x E h 1 − log E h
1
xh
log x h
0
1
log
1
xh
1
Eh
x
<E≤ xh
2
xh
0
1
log
1
xh
.
(27)
1
Eh
Now (see(4))
Mx0
x
<E≤ xh
2
xh
0
1
log
1
xh
1
Eh
≤ Mx0
1
Mx0
Mx0
1
=
E(x) ≤
(1 + )bh+1 x h+1
log 2
log 2
E≤x log 2
(28)
and (see (26))
1
1
1
hx h 1
hx h 1
hx h
1
1
(AE,h +). (29)
≤
≤
log x E≤ x E h1 1 − log E h1
log x E≤ x E h1 1 − log E h1
log x
1
1
2h
xh
0
log x h
log x h
Consequently (27), (28) and (29) give
⎛
G1 (x) = o ⎝
1
h
⎞
x ⎠
.
log x
(30)
We have (see (6) and (4))
F1 (x) = c1
E1 ≤x
1 + c2
E2 ≤x
1 + · · · + cs
1
Es ≤x
= c1 E1 (x) + c2 E2 (x) + · · · + cs Es (x)
≤ (c1 + c2 + · · · + cs )(E1 (x) + E2 (x) + · · · + Es (x))
1
≤ (c1 + c2 + · · · + cs )(1 + )bh+1 x h+1 .
Since the numbers of the form E1 , E2 , . . . , Es are (h+1)-ful numbers.
For example if Eph are the numbers of the form
p61 p62 p43 p44 p35 p36 p27 .
(31)
R. Jakimczuk
178
E1 will be the numbers of the form
p81 p62 p43 p44 p35 p36 ,
where c1 = 1.
E2 will be the numbers of the form
p61 p62 p63 p44 p35 p36 ,
where c2 = 3.
E3 will be the numbers of the form
p61 p62 p53 p44 p45 p36 ,
where c3 = 1. In this case s = 3.
Consequently (31) gives
⎛
1
⎞
xh ⎠
.
F1 (x) = o ⎝
log x
(32)
Finally, (6), (26), (30) and (32) give (5). The theorem is thus proved.
Corollary 2.2 The following asymptotic formula holds
1
An ∼ h nh logh n.
AE,h
(33)
Proof. From (5) we obtain
log A(x) ∼
1
log x.
h
That is,
log x ∼ h log A(x).
Substituting this equation into (5) we find that
1
x ∼ h A(x)h logh A(x).
AE,h
Substituting x = An into this equation we obtain (33). The corollary is proved.
References
[1] A. Ivic, The Riemann zeta-function, Dover, 2003.
[2] R. Jakimczuk, On the distribution of certain composite numbers, International Journal of Contemporary Mathematical Sciences, 3 (2008), 1245
- 1254.
Received: August, 2011