Gulf Journal of Mathematics
Vol 3, Issue 1 (2015) 29-35
THE LOGARITHM OF THE LEAST PRIME FACTOR II
RAFAEL JAKIMCZUK1
∗
Abstract. Let p(n) be the least prime
Pnfactor of the positive integer n. In a
previous note we prove the equation i=2 log p(i) = o(n log n). In this note
we obtain more precise results.
1. Preliminary Results
In this note pn denotes the n-th prime number and p denotes a positive prime.
Then p1 = 2, p2 = 3, p3 = 5, p4 = 7, p5 = 11, . . ..
Let P (n) be the greatest prime factor of the positive integer n. N. G. de Bruijn
[1] proved the following asymptotic formula
n
X
log P (i) = an log n + O(n),
i=2
where a is a positive constant defined in [1].
Let p(n) be the least prime factor of the positive integer n. Clearly we have
n
n
X
X
log p(i) ≤
log P (i).
i=2
i=2
In a previous note [3] we prove the following theorem
Theorem 1.1. We have
n
X
log p(i) = ng(n),
i=2
where g(n) → ∞ and g(n) = o(log n). Therefore
n
X
log p(i) = o(n log n).
i=2
Proof. See [3].
In this note we obtain more precise results.
We need the following theorems.
Date: Received: Nov 10, 2013; Accepted: Apr 1, 2014.
∗
Corresponding author.
2010 Mathematics Subject Classification. Primary 11A99; Secondary 11B99.
Key words and phrases. Least prime factor of a positive integer, logarithm.
29
30
RAFAEL JAKIMCZUK
The following theorem is sometimes called either the principle of inclusionexclusion or the principle of cross-classification. We now enunciate the principle.
Theorem 1.2. Let S be a set of N distinct elements, and let S1 , . . . , Sr be arbitrary subsets of S containing N1 , . . . , Nr elements, respectively. For 1 ≤ i <
j < . . . < l ≤ r, let Sij...l be the intersection of Si , Sj , . . . , Sl and let Nij...l be the
number of elements of Sij...l . Then the number K of elements of S not in any of
S1 , . . . , Sr is
X
X
X
K=N−
Ni +
Nij −
Nijk + · · · + (−1)r N12...r .
1≤i≤r
1≤i<j≤r
1≤i<j<k≤r
Proof. See, for example, either [2, page 233] or [4, page 84].
P∞
P∞
Theorem 1.3. Let
i=1 ai and
i=1 bi be two series of positive terms such that
P
limi→∞ abii = 1. If ∞
b
is
divergent
then we have the following limit
i=1 i
Pn
ai
= 1.
lim Pi=1
n
n→∞
i=1 bi
P
Therefore ∞
i=1 ai is also divergent.
Proof. See [5, page 332].
Theorem 1.4. (Mertens) The following asymptotic formula holds
Y
1
e−γ
1−
,
∼
p
log x
p≤x
where γ is Euler’s constant.
Proof. See [2, chapter XXII].
Theorem 1.5. The following asymptotic formula holds
X1
∼ log log x.
p
p≤x
Proof. See [2, chapter XXII].
Theorem 1.6. (Prime Number Theorem)Let π(x) be the prime counting function. We have
X
x
π(x) ∼
,
pn ∼ n log n,
log p ∼ x.
log x
p≤x
We also need the following well-known formulae,
k X
k k−i i
k
(a + b) =
a b,
i
i=0
20 + 21 + 22 + · · · + 2n = 2n+1 − 1.
In this note (as usual) b.c denotes the integer-part function. Note that
0 ≤ x − bxc < 1.
(1.1)
(1.2)
(1.3)
LOGARITHM OF THE LEAST PRIME FACTOR
31
2. Main results
Let βph (n) be the number of numbers that do not exceed n such that their least
prime factor is ph . Therefore
X
βph (n) = n − 1.
(2.1)
ph ≤n
We have the following theorem.
Theorem 2.1. The following asymptotic formula holds
!
h−1
Y
1
1
βph (n) =
1−
n + Oph (1)
p
p
i
h
i=1
(n ≥ 1),
where |Oph (1)| ≤ 2h−1 (n ≥ 1). Consequently these numbers have positive density
!
h−1
Y
1
1
.
Dph =
1−
pi
ph
i=1
Proof. We have (see Theorem 1.2)
X n X n X
n
n
βph (n) =
−
+
−
ph
p h pi
ph pi p j
p h pi pj pk
1≤i<j<k≤h−1
1≤i≤h−1
1≤i<j≤h−1
X
X
n
n
n
n
−
+
+ · · · + (−1)h−1
=
ph p1 · · · ph−1
ph 1≤i≤h−1 ph pi 1≤i<j≤h−1 ph pi pj
X
n
n
−
+ · · · + (−1)h−1
+ Oph (1)
p
p
h pi pj pk
h p1 · · · ph−1
1≤i<j<k≤h−1
!
h−1 n Y
1
+ Oph (1),
(2.2)
=
1−
ph i=1
pi
where (see (2.2), (1.3) and (1.1))
h−1 X
h−1
|Oph (1)| ≤
= (1 + 1)h−1 = 2h−1 .
i
i=0
Theorem 2.2. The following asymptotic formula holds
X
log ph βph (n) ∼ e−γ n log log log n.
ph ≤log n
Proof. We have (Theorem 2.1, Theorem 1.6 and Theorem 1.4)
Dph ∼
e−γ
.
ph log ph
That is
log ph Dph ∼
e−γ
.
ph
(2.3)
32
RAFAEL JAKIMCZUK
Hence (Theorem 1.3)
k
X
log ph Dph ∼
k
X
e−γ
h=1
h=1
ph
,
and theorefore (Theorem 1.5)
X
X e−γ
∼ e−γ log log n.
log ph Dph ∼
p
h
p ≤n
p ≤n
h
(2.4)
h
We have (Theorem 2.1)
X
X
log ph βph (n) =
(log ph Dph n + log ph Oph (1))
ph ≤log n
ph ≤log n
!
=
X
log ph Dph
X
n+
ph ≤log n
log ph Oph (1).
(2.5)
n ∼ e−γ n log log log n,
(2.6)
ph ≤log n
Now (see (2.4))
!
X
log ph Dph
ph ≤log n
and, on the other hand, we have (Theorem 2.1, Theorem 1.6 and (1.2))
X
X
X
2h−1
log ph |Oph (1)| ≤ log log n
log ph Oph (1) ≤
ph ≤log n
ph ≤log n
ph ≤log n
1−1
2−1
π(log n)−1
π(log n)
= log log n 2
+2
+ ··· + 2
= log log n 2
−1
log n
c log 2
≤ log log n2π(log n) ≤ log log nec log 2 log log n = n log log n log log n
√
≤
n log log n,
where c > 1. Equations (2.5), (2.6) and (2.7) give (2.3).
(2.7)
Note that the number of numbers not exceeding n such that their least prime
factor do not exceed log n considered in Theorem 2.2, namely
X
βph (n)
ph ≤log n
is not a small number, since we have (see [3] and equation (2.1))
X
βph (n) ∼ n.
ph ≤log n
In the following theorem we obtain a more precise result.
Theorem 2.3. The following asymptotic formulae hold
X
n
n
−γ
βph (n) = n − e
+o
,
log log n
log log n
ph ≤log n
X
n
βph (n) ∼ e−γ
.
log log n
log n<p ≤n
h
(2.8)
(2.9)
LOGARITHM OF THE LEAST PRIME FACTOR
33
Proof. The following equality can be proved without difficulty by mathematical
induction (see Theorem 2.1)
!
k
k
h−1
k X
X
Y
Y
1
1
1
Dph =
=1−
(k ≥ 1).
1−
1−
pi
ph
pi
i=1
i=1
h=1
h=1
Therefore (Theorem 1.4)
X
Y
1
e−γ
1
Dph = 1 −
1−
=1−
+o
.
p
log
n
log
n
h
p ≤n
p ≤n
h
(2.10)
h
We have (Theorem 2.1)
X
X
X
βph (n) = n
Dph +
Oph (1).
ph ≤log n
ph ≤log n
(2.11)
ph ≤log n
Now (Theorem 2.1, Theorem 1.6 and (1.2))
X
X
X
2h−1
|Oph (1)| ≤
Oph (1) ≤
= 2
1−1
+2
ph ≤log n
ph ≤log n
ph ≤log n
2−1
+ ··· + 2
π(log n)−1
log n
c log 2 log
log n
≤ 2π(log n) ≤ e
√
n.
≤
=n
π(log n)
=2
−1
c log 2
log log n
(2.12)
Equations (2.11), (2.10) and (2.12) give (2.8). Finally (2.9) is an immediate
consequence of (2.8).
Now, we can obtain a better theorem that Theorem 2.2.
Theorem 2.4. Let k be a positive integer. The following asymptotic formula
holds
X
log ph βph (n) ∼ e−γ n log log log n.
(2.13)
ph ≤logk n
Proof. We have
X
X
log ph βph (n) =
log ph βph (n) +
k
ph ≤log n
ph ≤log n
X
(2.14)
log n<ph ≤log n
On the other hand (see (2.9)) we have
X
log ph βph (n) ≤ log(logk n)
X
βph (n)
log n<ph ≤n
k
log n<ph ≤log n
∼ k log log ne−γ
log ph βph (n).
k
n
∼ ke−γ n.
log log n
Equations (2.14), (2.3) and (2.15) give (2.13).
Now, we can prove a better theorem that Theorem 1.1.
(2.15)
34
RAFAEL JAKIMCZUK
Theorem 2.5. The following inequality holds
n
X
X
log p(i) =
log ph βph (n) ≤ c
ph ≤n
i=2
1
n log n,
log log n
(2.16)
X
(2.17)
where c is a positive constant.
Proof. We have
X
X
log ph βph (n) =
log ph βph (n) +
ph ≤n
ph ≤log n
log n<ph ≤n
On the other hand (see (2.9)) we have
X
X
log ph βph (n) ≤ log n
log n<ph ≤n
log ph βph (n).
βph (n) ∼ log ne−γ
log n<ph ≤n
n
.
log log n
(2.18)
Equations (2.18), (2.3) and (2.17) give (2.16).
We establish the following conjecture.
n
X
log p(i) ∼ e−γ n log log n.
i=2
To finish, we have the following theorem.
Theorem 2.6. Let k ≥ 2 be an arbitrary but fixed positive integer. We have the
following asymptotic formulae
X
1
log ph βph (n) ∼ 1 −
n,
k
n
k
<ph ≤n
X
n
<ph ≤n
k
n
1
,
βph (n) ∼ 1 −
k log n
X
√
log ph βph (n) ∼ n,
n<ph ≤n
X
√
βph (n) ∼
n<ph ≤n
n
.
log n
√
Proof. Note that if n < ph ≤ n we have p2h > n. Hence there exists an unique
multiple of ph not exceeding n such that ph is its least prime factor, namely ph .
Therefore in the four formulae of the theorem we have
βph (n) = 1.
(2.19)
Now, the theorem is an immediate consequence of (2.19) and Theorem 1.6.
Acknowledgement. The author is very grateful to Universidad Nacional de
Luján.
LOGARITHM OF THE LEAST PRIME FACTOR
35
References
1. N. G. de Bruijn, On the number of positive integers ≤ x and free of prime factors > y,
Kon. Nederl. Akad. Wetensch. Proceedings A 54(= Indagationes Math. 13) (1951), 50 - 60.
2. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford, 1960.
3. R. Jakimczuk, The logarithm of the least prime factor, Gulf J. Math. 1(2013), 1-4.
4. W. J. LeVeque, Topics in Number Theory, Addison-Wesley, 1958.
5. J. Rey Pastor, P. Pi Calleja y C. A. Trejo, Análisis Matemático (Volumen 1), Kapelusz,
1969.
1
División Matemática, Universidad Nacional de Luján, Buenos Aires, Argentina.
E-mail address: [email protected]