Integers with a large smooth divisor

0.14 MB·English

by  William D. Banks

6 Integers with a large smooth divisor
0
0
2
William D. Banks
n
a
Department of Mathematics
J

9 University of Missouri
1
Columbia, MO 65211, USA

]
T [email protected]
N
Igor E. Shparlinski
.
h
t Department of Computing
a
m
Macquarie University
[
Sydney, NSW 2109, Australia

1
v [email protected]
0
6
4
1 Abstract
0
6
WestudythefunctionΘ(x,y,z)thatcountsthenumberofpositive
0
/ integersn ≤ xwhichhaveadivisord > z withthepropertythatp ≤ y
h
for every prime p dividing d. We also indicate some cryptographic
t
a
applications of our results.
m
:
v
i 1 Introduction
X
r
a For every integer n ≥ 2, let P+(n) and P−(n) denote the largest and the
smallest prime factor of n, respectively, and put P+(1) = 1, P−(1) = ∞. For
real numbers x,y ≥ 1, let Ψ(x,y) and Φ(x,y) denote the counting functions
of the sets of y-smooth numbers and y-rough numbers, respectively; that is,
Ψ(x,y) = #{n ≤ x : P+(n) ≤ y},
Φ(x,y) = #{n ≤ x : P−(n) > y}.
1
For a very wide range in the xy-plane, it is known that
x
Ψ(x,y) ∼ ̺(u)x and Φ(x,y) ∼ ω(u) ,
logy
where u denotes the ratio (logx)/logy, ̺(u) is the Dickman function, and
ω(u) is the Buchstab function; the definitions and certain analytic properties
of ̺(u) and ω(u) are reviewed in Sections 2 and 3 below.
In this paper, our principal object of study is the function Θ(x,y,z) that
counts positive integers n ≤ x for which there exists a divisor d | n with
d > z and P+(d) ≤ y; in other words,
Θ(x,y,z) = #{n ≤ x : n > z},
y
where n denotes the largest y-smooth divisor of n. The function Θ(x,y,z)
y
has been previously studied in the literature; see [1, 6, 7, 8].
For x,y,z varying over a wide domain, we derive the first two terms of
the asymptotic expansion of Θ(x,y,z). We show that the main term can be
naturally defined in terms of the partial convolution C (u,v) of ̺ with ω,
ω,̺
which is defined by
∞
C (u,v) = ω(u−s)̺(s)ds.
ω,̺
Zv
Using precise estimates for Ψ(x,y) and Φ(x,y), we also identify the second
term of the asymptotic expansion of Θ(x,y,z), which is naturally expressed
in terms of the partial convolution Cω,̺′(u,v) of ̺′ with ω:
∞
Cω,̺′(u,v) = ω(u−s)̺′(s)ds.
Zv
We use this formula to give a heuristic prediction for the density of certain
integers of cryptographic interest which appear in [3]. An alternative ap-
proach, which establishes a two term asymptotic formula for Θ(x,y,z) over
a wider range, has been developed recently by Tenenbaum [8].
Theorem 1. For fixed ε > 0 and uniformly in the domain
x ≥ 3, y ≥ exp{(loglogx)5/3+ε}, ylogy ≤ z ≤ x/y,
we have
x
Θ(x,y,z) = ̺(u)+Cω,̺(u,v) x−γCω,̺′(u,v) +O E(x,y,z) ,
logy
(cid:0) (cid:1) (cid:0) (cid:1)
2
where u = (logx)/logy, v = (logz)/logy, γ is the Euler-Mascheroni con-
stant, and
x ̺(v)log(v+1) ̺(v)
E(x,y,z) = ̺(u−1)+ + .
logy logy log(v +1)
(cid:26) (cid:27)
The proofofTheorem 1is given belowinSection 4; ourprincipal toolsare
the estimates of Lemma 4 (Section 2) and Lemma 6 (Section 3). In Section 5,
we outline some cryptographic applications of our results.
Acknowledgements. The authors would like to thank Alfred Menezes for
bringing to our attention the cryptographic applications which initially moti-
vated our work. We also thank G´eraldTenenbaum for pointing out a mistake
in the original manuscript, and for many subsequent discussions. This work
was started during a visit by W. B. to Macquarie University; the support and
hospitality of this institution are gratefully acknowledged. During the prepa-
ration of this paper, I. S. was supported in part by ARC grant DP0556431.
2 Integers free of large prime factors
In this section, we collect various estimates for the counting function Ψ(x,y)
of y-smooth numbers:
Ψ(x,y) = #{n ≤ x : P+(n) ≤ y}.
As usual, we denote by ̺(u) the Dickman function; it is continuous at u = 1,
differentiable for u > 1, and it satisfies the difference-differential equation
u̺′(u)+̺(u−1) = 0 (u > 1) (1)
along with the initial condition
̺(u) = 1 (0 ≤ u ≤ 1).
It is convenient to define ̺(u) = 0 for all u < 0 so that (1) is satisfied for
u ∈ R \ {0,1}, and we also define ̺′(u) by right-continuity at u = 0 and
u = 1. For a discussion of the analytic properties of ̺(u), we refer the reader
to [6, Chapter III.5].
We need the following well known estimate for Ψ(x,y), which is due to
Hildebrand [2] (see also [6, Corollary 9.3, Chapter III.5]):
3
Lemma 1. For fixed ε > 0 and uniformly in the domain
x ≥ 3, x ≥ y ≥ exp{(loglogx)5/3+ε},
we have
log(u+1)
Ψ(x,y) = ̺(u)x 1+O ,
logy
(cid:26) (cid:18) (cid:19)(cid:27)
where u = (logx)/logy.
We also need the following extension of Lemma 1, which is a special case
of the results of Saias [5]:
Lemma 2. For fixed ε > 0 and uniformly in the domain
x ≥ 3, y ≥ exp{(loglogx)5/3+ε}, x ≥ ylogy,
the following estimate holds:
x x
Ψ(x,y) = ̺(u)x+(γ −1)̺′(u) +O ̺′′(u) ,
logy log2y
(cid:18) (cid:19)
where u = (logx)/logy.
The following lemma provides a precise estimate for the sum
1
S(y,z) =
d
d>z
P+X(d)≤y
over a wide range, which is used in the proofs of Lemmas 4 and 6 below. The
sum S(y,z) has been previously studied; see, for example, [7].
Lemma 3. For fixed ε > 0 and uniformly in the domain
y ≥ 3, 1 ≤ z ≤ expexp{(logy)3/5−ε},
we have
S(y,z) = τ(v)logy −γ̺(v)+O E(y,z) ,
where v = (logz)/logy, (cid:0) (cid:1)
∞
τ(v) = ̺(s)ds,
Zv
4
and
̺(v)log(v +1)
if z ≥ ylogy;
logy
E(y,z) = 

 loglogy
 z−1 + if z < ylogy.
logy


Proof. Let Y = ylogy. First, suppose that z > Y, and put

exp{(logy)3/5−ε/2}
T = .
logy
By partial summation, it follows that
1
S(y,z) = +S(y,yT)
d
z<Xd≤yT
P+(d)≤y (2)
Ψ(yT,y) Ψ(z,y) T Ψ(ys,y)
= − +logy ds+S(y,yT).
yT z ys
Zv
By Lemma 1, we have the estimate
Ψ(z,y) ̺(v)log(v +1)
= ̺(v)+O .
z logy
(cid:18) (cid:19)
Also, by our choice of T we have
Ψ(yT,y) ̺(v)log(v +1)
≪ ̺(T) ≪ . (3)
yT logy
The following bound is given in the proof of [7, Corollaire 2]:
1
S(y,yT) = ≪ ̺(T)eεT +y−(1−ε)T,
d
dX>yT
P+(d)≤y
from which we deduce that
̺(v)log(v +1)
S(y,yT) ≪ . (4)
logy
To estimate the integral in (2), we apply Lemma 2 and write
T Ψ(ys,y)
ds = I +I +O(I ),
ys 1 2 3
Zv
5
where
T
I = ̺(s)ds = τ(v)−τ(T),
1
Zv
(γ −1) T (γ −1)(̺(T)−̺(v))
I = ̺′(s)ds = ,
2
logy logy
Zv
1 T ̺′(T)−̺′(v)
I = ̺′′(s)ds = .
3 log2y log2y
Zv
Since |̺′(v)| ≍ ̺(v)log(v +1), and
̺(v)log(v+1)
τ(T) ≪ ̺(T) ≪ ,
log2y
it follows that
T Ψ(ys,y) (γ −1)̺(v) ̺(v)log(v+1)
ds = τ(v)− +O . (5)
ys logy log2y
Zv (cid:18) (cid:19)
Inserting the estimates (3), (4) and (5) into (2), we obtain the desired esti-
mate in the case z > Y.
Next, suppose that y ≤ z ≤ Y, and put
logY loglogy
V = = 1+ .
logy logy
Since ̺(s) = 1−logs for 1 ≤ s ≤ 2, we have
loglogy
1 ≥ ̺(v) ≥ ̺(V) = 1+O ;
logy
(cid:18) (cid:19)
therefore,
loglogy
̺(v)−̺(V) ≪ . (6)
logy
By partial summation, it follows that
1
S(y,z) = +S(y,Y)
d
z<d≤Y
X
P+(d)≤y (7)
Ψ(Y,y) Ψ(z,y) V Ψ(ys,y)
= − +logy ds+S(y,Y).
Y z ys
Zv
6
Using Lemma 1 together with (6), it follows that
Ψ(Y,y) Ψ(z,y) 1 loglogy
− = ̺(V)−̺(v)+O ≪ . (8)
Y z logy logy
(cid:18) (cid:19)
Applying the estimate from the previous case, we also have
1
S(y,Y) = τ(V)logY −γ̺(V)+O . (9)
logy
(cid:18) (cid:19)
To estimate the integral in (7), we use Lemma 1 again and write
V Ψ(ys,y)
ds = I +O(I ),
ys 4 5
Zv
where
V
I = ̺(s)ds = τ(v)−τ(V),
4
Zv
1 V log(Y/z) loglogy
I = ds = ≪ .
5 logy log2y log2y
Zv
Therefore,
V Ψ(ys,y) loglogy
ds = τ(v)−τ(V)+O . (10)
ys log2y
Zv (cid:18) (cid:19)
Inserting the estimates (8), (9) and (10) into (7), and taking into account (6),
we obtain the stated estimate for y ≤ z ≤ Y.
Finally, suppose that 1 ≤ z < y. In this case,
1
S(y,z) = +S(y,y). (11)
d
z<d≤y
X
By partial summation, we have
1
= logy −logz +O(z−1) = (1−v)logy +O(z−1)
d
z<d≤y
X
1
= logy ̺(s)ds+O(z−1) = (τ(v)−τ(1))logy +O(z−1).
Zv
7
Applying the estimate from the previous case, we also have
loglogy
S(y,y) = τ(1)logy −γ̺(1)+O .
logy
(cid:18) (cid:19)
Inserting these estimates into (11), and using the fact that ̺(v) = ̺(1) = 1,
we obtain the desired result. This completes the proof.
Lemma 4. For fixed ε > 0 and uniformly in the domain
x ≥ 3, y ≥ exp{(loglogx)5/3+ε}, 1 ≤ z ≤ x/y,
we have
̺(u−u )
d
≪ C (u,v)log(u+1)+̺(u−v)̺(v)+̺(u−1),
̺,̺
d
z<d≤x/y
X
P+(d)≤y
where u = (logx)/logy, v = (logz)/logy, u = (logd)/logy for every
d
integer d in the sum, and
∞
C (u,v) = ̺(u−s)̺(s)ds.
̺,̺
Zv
Proof. By partial summation, we have
̺(u−u ) u−1
d = S(y,x/y)−̺(u−v)S(y,z)+ ̺′(u−s)S(y,ys)ds.
d
z<d≤x/y Zv
X
P+(d)≤y
Lemma 3 implies that
S(y,x/y) = τ(u−1)logy +O ̺(u−1) ,
S(y,z) = τ(v)logy +O ̺(v) ,
(cid:0) (cid:1)
and (cid:0) (cid:1)
u−1
̺′(u−s)S(y,ys)ds = I logy +O(I ),
1 2
Zv
where
u−1
I = ̺′(u−s)τ(s)ds = ̺(u−v)τ(v)−τ(u−1)+C (u,v),
1 ̺,̺
Zv
u−1
I = ̺′(u−s) ̺(s)ds.
2
Zv
(cid:12) (cid:12)
(cid:12) (cid:12)
8
Finally, using the bound
̺′(t) ≪ ̺(t)log(t+1) (t > 1),
we see that (cid:12) (cid:12)
(cid:12) (cid:12)
u−1
I ≪ log(u+1) ̺(u−s)̺(s)ds ≤ C (u,v)log(u+1).
2 ̺,̺
Zv
Putting everything together, the result follows.
3 Integers free of small prime factors
In this section, we collect various estimates for the counting function Φ(x,y)
of y-rough numbers:
Φ(x,y) = #{n ≤ x : P−(n) > y}.
Asusual, we denoteby ω(u)theBuchstab function; foru > 1, it istheunique
continuous solution to the difference-differential equation
′
uω(u) = ω(u−1) (u > 2) (12)
with initial condition (cid:0) (cid:1)
uω(u) = 1 (1 ≤ u ≤ 2).
It is convenient to define ω(u) = 0 for all u < 1 so that (12) is satisfied for
u ∈ R \ {1,2}, and we also define ω′(u) by right-continuity at u = 1 and
u = 2. For a discussion of the analytic properties of ω(u), we refer the reader
to [6, Chapter III.6]
The next result follows from [6, Corollary 7.5, Chapter III.6]:
Lemma 5. For fixed ε > 0 and uniformly in the domain
x ≥ 3, x ≥ y ≥ exp{(loglogx)5/3+ε},
the following estimate holds:
eγ x̺(u)
Φ(x,y) = xω(u)−y +O ,
ζ(1,y) log2y
(cid:18) (cid:19)
(cid:0) (cid:1)
where u = (logx)/logy, and ζ(1,y) = (1−p−1)−1.
p≤y
Q
9
Lemma 6. For fixed ε > 0 and uniformly in the domain
x ≥ 3, y ≥ exp{(loglogx)5/3+ε}, 1 ≤ z ≤ x/y,
we have
ω(u−u )
d
= Cω,̺(u,v)logy −γCω,̺′(u,v)+O E(y,z) ,
d
z<d≤x/y
P+X(d)≤y (cid:0) (cid:1)
where u = (logx)/logy, v = (logz)/logy, u = (logd)/logy for every
d
integer d in the sum, and E(y,z) is the error term of Lemma 3.
Proof. By partial summation, it follows that
ω(u−u ) u−1
d = S(y,x/y)−ω(u−v)S(y,z)+ ω′(u−s)S(y,ys)ds.
d
z<d≤x/y Zv
X
P+(d)≤y
By Lemma 3 we have the estimates
S(y,x/y) = τ(u−1)logy −γ̺(u−1)+O E(y,x/y)
and (cid:0) (cid:1)
S(y,z) = τ(v)logy −γ̺(v)+O E(y,z) .
Also, (cid:0) (cid:1)
u−1
ω′(u−s)S(y,ys)ds = I logy −γI +O(I ),
1 2 3
Zv
where
u−1
I = ω′(u−s)τ(s)ds = ω(u−v)τ(v)−τ(u−1)+C (u,v),
1 ω,̺
Zv
u−1
I2 = ω′(u−s)̺(s)ds = ω(u−v)̺(v)−̺(u−1)+Cω,̺′(u,v),
Zv
1 u−1
I = ω′(u−s) E(y,ys)ds.
3
logy
Zv
(cid:12) (cid:12)
Putting everything to(cid:12)gether, w(cid:12)e see that the stated estimate follows from the
bound
E(y,x/y)+ω(u−v)E(y,z)+I ≪ E(y,z). (13)
3
10