INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A17

INTEGERS WITH A LARGE SMOOTH DIVISOR

William D. Banks

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

                          bbanks@math.missouri.edu

Igor E. Shparlinski

Department of Computing, Macquarie University, Sydney, NSW 2109, Australia

                            igor@ics.mq.edu.au

Received: 7/19/06, Revised: 1/31/07, Accepted: 2/2/07, Published: 4/3/07

Abstract

We study the function Θ(x, y, z) that counts the number of positive integers n ≤ x which have a divisor d > z with the property that p ≤ y for every prime p dividing d. We also indicate some cryptographic applications of our results.

1. Introduction

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} and Φ(x, y) = #{n ≤ x : P^{−}(n) > y}.

For a very wide range in the xy-plane, it is known that

Ψ(x, y) ∼ ϱ(u) x and Φ(x, y) ∼ ω(u) \frac{x}{log y} ,

where u denotes the ratio (log x)/ log y, ϱ(u) is the Dickman function, and ω(u) is the Buchstab function; the definitions and certain analytic properties of ϱ(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_{y} > z},

where n_{y} denotes the largest y-smooth divisor of n. The function Θ(x, y, z) 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_{ω,ϱ}(u, v) of ϱ with ω, which is defined by

C_{ω,ϱ}(u, v) = ∫_{v}^{∞} ω(u − s)ϱ(s) ds.

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 C_{ω,ϱ′}(u, v) of ϱ′ with ω:

C_{ω,ϱ′}(u, v) = ∫_{v}^{∞} ω(u − s)ϱ′(s) ds.

Theorem 1. For fixed ε > 0 and uniformly in the domain

x ≥ 3,  y ≥ exp{(log log x)^{5/3+ε}},  y log y ≤ z ≤ x/y,

we have

Θ(x, y, z) = (ϱ(u) + C_{ω,ϱ}(u, v))x − γ C_{ω,ϱ′}(u, v) \frac{x}{log y} + O(E(x, y, z)),

where u = (log x)/ log y, v = (log z)/ log y, γ is the Euler-Mascheroni constant, and

E(x, y, z) = \frac{x}{log y} \{ϱ(u − 1) + \frac{ϱ(v) log(v + 1)}{log y} + \frac{ϱ(v)}{log(v + 1)}\} .

Similar results have been obtained concomitantly, using a more elaborate approach, by Tenenbaum [8].

The proof of Theorem 1 is given below in Section 4; our principal tools are the estimates of Lemma 4 (Section 2) and Lemma 6 (Section 3). In Section 5, we use the formula of Theorem 1 to give a heuristic prediction for the density of certain integers of cryptographic interest which appear in a work by Menezes [3].

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 ϱ(u) the Dickman function; it is continuous at u = 1, differentiable for u > 1, and it satisfies the differential-difference equation

uϱ′(u) + ϱ(u − 1) = 0 (u > 1),  (1)

along with the initial condition ϱ(u) = 1 (0 ≤ u ≤ 1). It is convenient to define ϱ(u) = 0 for all u < 0 so that (1) is satisfied for u ∈ ℝ\{0, 1}, and we also define ϱ′(u) by right-continuity at u = 0 and u = 1. For a discussion of the analytic properties of ϱ(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]):

Lemma 1. For fixed ε > 0 and uniformly in the domain

x ≥ 3, x ≥ y ≥ exp{(log log x)^{5/3+ε}},

we have

Ψ(x, y) = ϱ(u) x \{1 + O (\frac{log(u + 1)}{log y})\} ,

where u = (log x)/ log y.

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{(log log x)^{5/3+ε}},  x ≥ y log y,

the following estimate holds:

Ψ(x, y) = ϱ(u) x + (γ − 1)ϱ′(u) \frac{x}{log y} + O (ϱ′′(u)\frac{x}{log^{2} y} ) ,

where u = (log x)/ log y.

The following lemma provides a precise estimate for the sum

S(y, z) = ∑_{\begin{matrix}d>z \\ P^{+}(d)≤y \end{matrix}} \frac{1}{d}

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].

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Lemma 3. For fixed ε > 0 and uniformly in the domain

y ≥ 3, 1 ≤ z ≤ exp exp{(log y)^{3/5−ε}},

we have

S(y, z) = τ(v) log y − γϱ(v) + O(E(y, z)),

where v = (log z)/ log y,

τ(v) = ∫_{v}^{∞} ϱ(s) ds,

and

E(y, z) = \{ \begin{matrix}\frac{ϱ(v) log(v + 1)}{log y} & if z ≥ y log y; \\ \frac{1}{z} + \frac{log log y}{log y} & if z < y log y. \end{matrix}

Proof. Let Y = y log y. First, suppose that z > Y, and put T = \frac{exp\{(log y)^{3/5−ε/2}\}}{log y}. By partial summation, it follows that

\begin{aligned}S(y, z) & = ∑_{\begin{matrix}z<d≤y^{T} \\ P^{+}(d)≤y \end{matrix}} \frac{1}{d} + S(y, y^{T}) \\ & = \frac{Ψ(y^{T} , y)}{y^{T}} − \frac{Ψ(z, y)}{z} + log y ∫_{v}^{T} \frac{Ψ(y^{s}, y)}{y^{s}} ds + S(y, y^{T}). \end{aligned}  (2)

By Lemma 1, we have the estimate \frac{Ψ(z, y)}{z} = ϱ(v) + O (\frac{ϱ(v) log(v + 1)}{log y}) .

We now recall that

ϱ(w) = exp (−w log w + O(w log log w)) ;  (3)

see [5, Lemma 3(iv)]. Thus, by our choice of T we have

ϱ(T) = exp \{−(1 + o(1))\frac{exp\{(log y)^{3/5−ε/2}\}}{(log y)^{2/5+ε/2}}\} .

On the other hand, since v ≤ log z ≤ exp{(log y)^{3/5−ε}}, it follows that

\frac{ϱ(v) log(v + 1)}{log y} = \frac{exp (−v log v + O(v log log v))}{log y}
≥ exp {−(1 + o(1)) exp{(log y)^{3/5−ε}}(log y)^{3/5−ε}} .

Therefore,

\frac{Ψ(y^{T} , y)}{y^{T}} ≪ ϱ(T) ≪ \frac{ϱ(v) log(v + 1)}{log y}.  (4)

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The following bound is given in the proof of [7, Corollary 2]:

S(y, y^{T}) = ∑_{\begin{matrix}d>y^{T} \\ P^{+}(d)≤y \end{matrix}} \frac{1}{d} ≪ ϱ(T)e^{εT} + y^{−(1−ε)T} ,

from which we deduce that

S(y, y^{T}) ≪ \frac{ϱ(v) log(v + 1)}{log y}.  (5)

To estimate the integral in (2), we apply Lemma 2 and write

∫_{v}^{T} \frac{Ψ(y^{s}, y)}{y^{s}} ds = I_{1} + I_{2} + O(I_{3}),

where

I_{1} = ∫_{v}^{T} ϱ(s) ds = τ(v) − τ(T),
I_{2} = \frac{(γ − 1)}{log y} ∫_{v}^{T} ϱ′(s) ds = \frac{(γ − 1)(ϱ(T) − ϱ(v))}{log y},
I_{3} = \frac{1}{log^{2} y} ∫_{v}^{T} ϱ′′(s) ds = \frac{ϱ′(T) − ϱ′(v)}{log^{2} y}.

Using (1) together with [6, Lemma 8.1 and bound (61), Chapter III.5] we see that |ϱ′(v)| ≍ ϱ(v) log(v + 1). Furthermore, as in our derivation of (4), we see that (3) yields

τ(T) ≪ ϱ(T) ≪ \frac{ϱ(v) log(v + 1)}{log^{2} y}.

Therefore,

∫_{v}^{T} \frac{Ψ(y^{s}, y)}{y^{s}} ds = τ(v) − \frac{(γ − 1)ϱ(v)}{log y} + O (\frac{ϱ(v) log(v + 1)}{log^{2} y}) .  (6)

Inserting the estimates (4), (5), and (6) into (2), we obtain the desired estimate when z > Y.

Next, suppose that y ≤ z ≤ Y, and put V = \frac{log Y}{log y} = 1 + \frac{log log y}{log y}. Since ϱ(s) = 1 − log s for 1 ≤ s ≤ 2, we have 1 ≥ ϱ(v) ≥ ϱ(V) = 1 + O (\frac{log log y}{log y} ); therefore,

ϱ(v) − ϱ(V) ≪ \frac{log log y}{log y} .  (7)

By partial summation, it follows that

\begin{aligned}S(y, z) & = ∑_{\begin{matrix}z<d≤Y \\ P^{+}(d)≤y \end{matrix}} \frac{1}{d} + S(y, Y) \\ & = \frac{Ψ(Y, y)}{Y} − \frac{Ψ(z, y)}{z} + log y ∫_{v}^{V} \frac{Ψ(y^{s}, y)}{y^{s}} ds + S(y, Y). \end{aligned}  (8)

Using Lemma 1 together with (7), it follows that

\frac{Ψ(Y, y)}{Y} − \frac{Ψ(z, y)}{z} = ϱ(V) − ϱ(v) + O (\frac{1}{log y} ) ≪ \frac{log log y}{log y} .  (9)

Applying the estimate from the previous case, we also have

S(y, Y) = τ(V) log Y − γϱ(V) + O (\frac{1}{log y} ) .  (10)

To estimate the integral in (8), we use Lemma 1 again and write

∫_{v}^{V} \frac{Ψ(y^{s}, y)}{y^{s}} ds = I_{4} + O(I_{5}),

where

I_{4} = ∫_{v}^{V} ϱ(s) ds = τ(v) − τ(V),
I_{5} = \frac{1}{log y} ∫_{v}^{V} ds = \frac{log(Y/z)}{log^{2} y} ≪ \frac{log log y}{log^{2} y} .

Therefore,

∫_{v}^{V} \frac{Ψ(y^{s}, y)}{y^{s}} ds = τ(v) − τ(V) + O (\frac{log log y}{log^{2} y} ) .  (11)

Inserting the estimates (9), (10) and (11) into (8), and taking into account (7), we obtain the stated estimate for y ≤ z ≤ Y.

Finally, suppose that 1 ≤ z < y. In this case,

S(y, z) = ∑_{z<d≤y} \frac{1}{d} + S(y, y).  (12)

By partial summation, we have

∑_{z<d≤y} \frac{1}{d} = log y − log z + O(z^{−1}) = (1 − v) log y + O(z^{−1})

= log y ∫_{v}^{1} ϱ(s) ds + O(z^{−1}) = (τ(v) − τ(1)) log y + O(z^{−1}).

Applying the estimate from the previous case, we also have

S(y, y) = τ(1) log y − γϱ(1) + O (\frac{log log y}{log y} ) .

Inserting these estimates into (12), and using the fact that ϱ(v) = ϱ(1) = 1, we obtain the desired result.

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Lemma 4. For fixed ε > 0 and uniformly in the domain

x ≥ 3, y ≥ exp{(log log x)^{5/3+ε}}, 1 ≤ z ≤ x/y,

we have

∑_{\begin{matrix}z<d≤x/y \\ P^{+}(d)≤y \end{matrix}} \frac{ϱ(u − u_{d})}{d} ≪ C_{ϱ,ϱ}(u, v) log(u + 1) + ϱ(u − v)ϱ(v) + ϱ(u − 1),

where u = (log x)/ log y, v = (log z)/ log y, u_{d} = (log d)/ log y for every integer d in the sum, and

C_{ϱ,ϱ}(u, v) = ∫_{v}^{∞} ϱ(u − s)ϱ(s) ds.

Proof. By partial summation, we have

∑_{\begin{matrix}z<d≤x/y \\ P^{+}(d)≤y \end{matrix}} \frac{ϱ(u − u_{d})}{d} = S(y, x/y) − ϱ(u − v)S(y, z) + ∫_{v}^{u−1} ϱ′(u − s)S(y, y^{s}) ds.

Lemma 3 implies that

\begin{aligned}S(y, x/y) & = τ(u − 1) log y + O(ϱ(u − 1)), \\ S(y, z) & = τ(v) log y + O(ϱ(v)), \end{aligned}

and

∫_{v}^{u−1} ϱ′(u − s)S(y, y^{s}) ds = I_{1} log y + O(I_{2}),

where

\begin{aligned}I_{1} & = ∫_{v}^{u−1} ϱ′(u − s)τ(s) ds = ϱ(u − v)τ(v) − τ(u − 1) + C_{ϱ,ϱ}(u, v), \\ I_{2} & = ∫_{v}^{u−1}|ϱ′(u − s)|ϱ(s) ds. \end{aligned}

Finally, using the bound |ϱ′(t)| ≪ ϱ(t) log(t + 1) (for t > 1), we see that

I_{2} ≪ log(u + 1) ∫_{v}^{u−1} ϱ(u − s)ϱ(s) ds ≤ C_{ϱ,ϱ}(u, v) log(u + 1).

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}.

As usual, we denote by ω(u) the Buchstab function; for u > 1, it is the unique continuous solution to the differential-difference equation

(uω(u))′ = ω(u − 1) (u > 2)  (13)

with initial condition uω(u) = 1 (1 ≤ u ≤ 2). It is convenient to define ω(u) = 0 for all u < 1 so that (13) 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{(log log x)^{5/3+ε}},

the following estimate holds:

Φ(x, y) = (xω(u) − y)\frac{e^{γ}}{ζ(1, y)} + O (\frac{xϱ(u)}{log^{2} y}) ,

where u = (log x)/ log y, and ζ(1, y) = ∏_{p≤y} (1 − p^{−1})^{−1}.

Lemma 6. For fixed ε > 0 and uniformly in the domain

x ≥ 3,  y ≥ exp{(log log x)^{5/3+ε}},  1 ≤ z ≤ x/y,

we have

∑_{\begin{matrix}z<d≤x/y \\ P^{+}(d)≤y \end{matrix}} \frac{ω(u − u_{d})}{d} = C_{ω,ϱ}(u, v) log y − γ C_{ω,ϱ′}(u, v) + O(E(y, z)),

where u = (log x)/ log y, v = (log z)/ log y, u_{d} = (log d)/ log y for every integer d in the sum, and E(y, z) is the error term of Lemma 3.

Proof. By partial summation, it follows that

∑_{\begin{matrix}z<d≤x/y \\ P^{+}(d)≤y \end{matrix}} \frac{ω(u − u_{d})}{d} = S(y, x/y) − ω(u − v)S(y, z) + ∫_{v}^{u−1} ω′(u − s)S(y, y^{s}) ds.

By Lemma 3 we have the estimates S(y, x/y) = τ(u − 1) log y − γϱ(u − 1) + O(E(y, x/y)) and S(y, z) = τ(v) log y − γϱ(v) + O(E(y, z)). Also,

∫_{v}^{u−1} ω′(u − s)S(y, y^{s}) ds = I_{1} log y − γI_{2} + O(I_{3}),

where

\begin{aligned}I_{1} & = ∫_{v}^{u−1} ω′(u − s)τ(s) ds = ω(u − v)τ(v) − τ(u − 1) + C_{ω,ϱ}(u, v), \\ I_{2} & = ∫_{v}^{u−1} ω′(u − s)ϱ(s) ds = ω(u − v)ϱ(v) − ϱ(u − 1) + C_{ω,ϱ′}(u, v), \\ I_{3} & = \frac{1}{log y} ∫_{v}^{u−1}|ω′(u − s)|E(y, y^{s}) ds. \end{aligned}

Putting everything together, we see that the stated estimate follows from the bound

E(y, x/y) + ω(u − v)E(y, z) + I_{3} ≪ E(y, z).  (14)

To prove this, observe that E(y, z_{1}) ≪ E(y, z_{2}) holds for all z_{1} ≥ z_{2} ≥ 1. Therefore, E(y, x/y) ≪ E(y, z), and

I_{3} ≪ \frac{E(y, z)}{log y} ∫_{v}^{u−1}|ω′(u − s)| ds ≪ \frac{E(y, z)}{log y}.

Taking into account the fact that ω(u − v) ≍ 1, we derive (14), completing the proof.

4. Proof of Theorem 1

For fixed y, every positive integer n can be uniquely decomposed as a product n = de, where P^{+}(d) ≤ y and P^{−}(e) > y. Therefore,

Θ(x, y, z) = ∑_{\begin{matrix}z<d≤x \\ P^{+}(d)≤y \end{matrix}} ∑_{\begin{matrix}e≤x/d \\ P^{−}(e)>y \end{matrix}} 1 = ∑_{\begin{matrix}z<d≤x \\ P^{+}(d)≤y \end{matrix}} Φ(x/d, y)

= Ψ(x, y) − Ψ(x/y, y) + ∑_{\begin{matrix}z<d≤x/y \\ P^{+}(d)≤y \end{matrix}} Φ(x/d, y).

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Using Lemma 1, it follows that Ψ(x, y)−Ψ(x/y, y) = ϱ(u) x+O (\frac{ϱ(u − 1) x}{log y}) . By Lemma 5, we also have

∑_{\begin{matrix}z<d≤x/y \\ P^{+}(d)≤y \end{matrix}} Φ(x/d, y) = ∑_{\begin{matrix}z<d≤x/y \\ P^{+}(d)≤y \end{matrix}} \{(\frac{x ω(u − u_{d})}{d} − y) \frac{e^{γ}}{ζ(1, y)} + O (\frac{xϱ(u − u_{d})}{d log^{2} y})\}

= \frac{e^{γ}x}{ζ(1, y)} ∑_{\begin{matrix}z<d≤x/y \\ P^{+}(d)≤y \end{matrix}} \frac{ω(u − u_{d})}{d} − \frac{e^{γ}y}{ζ(1, y)}\{Ψ(x/y, y) − Ψ(z, y)\}

+ O(\frac{x}{log^{2} y} ∑_{\begin{matrix}z<d≤x/y \\ P^{+}(d)≤y \end{matrix}} \frac{ϱ(u − u_{d})}{d}).

(15)

Applying Lemma 1 again, we have −\frac{e^{γ}y}{ζ(1, y)}\{Ψ(x/y, y) − Ψ(z, y)\} ≪ \frac{ϱ(u − 1) x}{log y}. Inserting the estimates of Lemmas 4 and 6 into (15), and making use of the trivial estimate

C_{ϱ,ϱ}(u, v) log(u + 1) ≪ log y ∫_{v}^{∞} ϱ(s) ds ≪ \frac{ϱ(v) log y}{log(v + 1)}.

it is easy to see that

Θ(x, y, z) = (ϱ(u) + C_{ω,ϱ}(u, v)\frac{e^{γ} log y}{ζ(1, y)} )x − γ C_{ω,ϱ′}(u, v)\frac{e^{γ}x}{ζ(1, y)} + O(E(x, y, z)).

To complete to proof, we use the estimate (see Vinogradov [9]):

ζ(1, y) = e^{γ} log y(1 + exp{−c(log y)^{3/5}}),

which holds for some absolute constant c > 0, together with the trivial estimate

max\{C_{ω,ϱ}(u, v), C_{ω,ϱ′}(u, v)\} ≪ ∫_{v}^{∞} ϱ(s) ds ≪ \frac{ϱ(v)}{log(v + 1)}.

5. Cryptographic Applications Suppose that two primes p and q are selected for use in the Digital Signature Algorithm (see, for example, [4]) using the following standard method:

• Select a random m-bit prime q;

• Randomly generate k-bit integers n until a prime p = 2nq + 1 is reached.

The large subgroup attack described in [3, Section 3.2.3] leads one naturally to consider the following question: What is the probability η(k, ℓ, m) that n has a divisor s > q which is 2^{ℓ}-smooth?

It is natural to expect that the proportion of those integers in the set {2^{k−1} ≤ n < 2^{k}} having a large smooth divisor should be roughly the same as the proportion of integers in

{2^{k−1} ≤ n < 2^{k} : n = (p − 1)/(2q) for some prime p ≡ 1 (mod 2q)}

having a large smooth divisor. Accordingly, we expect that the probability η(k, ℓ, m) is reasonably close to

\frac{Θ(2^{k}, 2^{ℓ}, 2^{m}) − Θ(2^{k−1}, 2^{ℓ}, 2^{m})}{2^{k−1}}.

Theorem 1 then suggests that η(k, ℓ, m) ≈ 2 ℘(k, ℓ, m) − ℘(k − 1, ℓ, m), where

℘(k, ℓ, m) = ϱ(k/ℓ) + C_{ω,ϱ}(k/ℓ, m/ℓ) − \frac{γ C_{ω,ϱ′}(k/ℓ, m/ℓ)}{ℓ log 2}.

In particular, the most interesting choice of parameters at the present time is k = 863, ℓ = 80, and m = 160 (which produce a 1024-bit prime p); we expect η(863, 80, 160) ≈ 0.09576.

Acknowledgements. The authors would like to thank Alfred Menezes for bringing to our attention the cryptographic applications which initially motivated our work. We also thank Gérald Tenenbaum 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 preparation of this paper, I. S. was supported in part by ARC grant DP0556431.

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[6] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, Cambridge, UK, 1995.

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[8] G. Tenenbaum, Integers with a large friable component, Preprint, 2006.

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