Numbers with a large prime factor by

LXXXIX.2 (1999)

Numbers with a large prime factor

by

Hong-Quan Liu (Harbin) and Jie Wu (Nancy)

1. Introduction. Let P (x) be the greatest prime factor of the integer Q

x<n≤x+x^1/2

n. It is expected that P (x) ≥ x for x ≥ 2. However, this in- equality seems extremely difficult to verify. In 1969, Ramachandra [20, I]

obtained a non-trivial lower bound: P (x) ≥ x

for sufficiently large x.

This result has been improved consecutively by many authors. The best es- timate known to date is very far from the expected result. The historical records are as follows:

P (x) ≥ x

by Ramachandra [20, II], P (x) ≥ x

by Graham [8],

P (x) ≥ x

by Jia [16, I], P (x) ≥ x

by Baker [1], P (x) ≥ x

by Jia [16, II],

P (x) ≥ x

by Jia [16, III] and Liu [18], P (x) ≥ x

by Jia [16, IV],

P (x) ≥ x

by Baker and Harman [2].

We note that the last two papers are independent. In both, the same es- timates for exponential sums were used. But Baker and Harman [2] intro- duced the alternative sieve procedure, developed by Harman [10] and by Baker, Harman and Rivat [3], to get a better exponent. In this paper we shall prove a sharper lower bound.

Theorem 1. We have P (x) ≥ x

for sufficiently large x.

As Baker and Harman indicated in [2], it is very difficult to make any progress without new exponential sum estimates. Naturally we first treat

1991 Mathematics Subject Classification: Primary 11N05; Secondary 11L07, 11N36.

Research of H. Q. Liu supported by Harbin Institute of Technology and the natural science foundation of China.

[163]

the corresponding exponential sums S

:= X

h∼H

X

m∼M

X

n∼N

b

e

 xh mn

 ,

S

:= X

h∼H

X

m∼M

X

n∼N

a

b

e

 xh mn

 ,

where e(t) := e

, |a

| ≤ 1, |b

| ≤ 1 and m ∼ M means cM < m ≤ c

M with some positive unspecified constants c, c

. The improvement in The- orem 1 comes principally from our new bound for S

(§2, Corollary 2), where we extend the condition N ≤ x

of Jia [16, III] and Liu [18]

to N ≤ x

(ε is an arbitrarily small positive number). It is noteworthy that we prove this as an immediate consequence of a new estimate on special bilinear exponential sums (§2, Theorem 2). This estimate has other applica- tions, which will be taken up elsewhere. Our results on S

(§3, Theorem 3) improve Theorem 6 of [7] (or [18], Lemma 2) and Lemma 14 of [1]. We need Lemma 9 of [1] only in a very short interval (3/5 ≤ θ ≤ 11/18).

If the interval (x, x + x

] is replaced by (x, x + x

], one can do much better. In 1973, Jutila [17] proved that the largest prime factor of Q

x<n≤x+x^1/2+ε

n is at least x

for x ≥ x

(ε). The exponent 2/3 was improved successively to 0.73 by Balog [4, I], to 0.772 by Balog [4, II], to 0.82 by Balog, Harman and Pintz [5], to 11/12 by Heath-Brown [12] and to 17/18 by Heath-Brown and Jia [13]. It should be noted that their methods cannot be applied to treat P (x), and this leads to the comparative weakness of the results on P (x) (cf. [5]).

Throughout this paper, we put L := log x, y := x

, N (d) := |{x < n ≤ x + y : d | n}| and v := x

. From [16, III], [18] and [2], in order to prove Theorem 1 it is sufficient to show

(1.1) X

x^0.6−ε<p≤x^0.738

N (p) log p < 0.4 yL,

where p denotes a prime number. For this we shall need an upper bound for the quantity S(θ) := P

x^θ<p≤ex^θ

N (p) (0.6 ≤ θ ≤ 0.738). We write

(1.2) S(θ) = X

x^θ<p≤ex^θ

X

x<mp≤x+y

1 = X

p∈A

1 = S(A, (ev)

),

where A = A(θ) := {n : x

< n ≤ ex

, N (n) = 1}, S(A, z) := |{n ∈ A : P

(n) ≥ z}| and P

(n) := min

p (P

(1) = ∞). We would like to give an upper bound for S(θ) of the form

(1.3) S(θ) ≤ {1 + O(ε)} u(θ)y

θL ,

where u(θ) is as small as possible. Thus in order to prove (1.1), it suffices to

show (1.3) and (1.4)

0.738

\

0.6

u(θ) dθ < 0.4.

As in [2], we shall prove (1.3) by the alternative sieve for 0.6 ≤ θ ≤ 0.661 and by the Rosser–Iwaniec sieve for 0.661 ≤ θ ≤ 0.738. Thanks to our new estimates for exponential sums, our u(θ) is strictly smaller than that of Baker and Harman [2].

In the sequel, we use ε

to denote a suitably small positive number, ε an arbitrarily small positive number, ε

an unspecified constant multiple of ε and put η := e

.

2. Estimates for bilinear exponential sums and for S

. First we investigate a special bilinear sum of type II:

S(M, N ) := X

m∼M

X

n∼N

a

b

e



X m

n

M

N

 .

Here the exponent 1/2 is important in our method. We have the following result.

Theorem 2. Let β ∈ R with β(β − 1) 6= 0, X > 0, M ≥ 1, N ≥ 1, L

:= log(2 + XM N ), |a

| ≤ 1 and |b

| ≤ 1. Then

S(M, N )  {(X

M

N

)

+ (X

M

N

)

+ (X

M

N

)

+ (XM

N

)

+ M N

+ M

N + X

M N }L

. P r o o f. In view of Theorem 2 of [7] (or Lemma 3.1 below), we can suppose X ≥ N . In addition we may also assume β > 0. Let Q ∈ (0, ε

N ] be a parameter to be chosen later. By the Cauchy–Schwarz inequality and Lemma 2.5 of [9], we have

|S|

 (M N )

Q + M

N

Q

X

1≤|q₁|<Q



1 − |q

| Q

 X

n∼N

b

b

X

m∼M

m

e(Am

t),

where t = t(n, q

) := (n + q

)

− n

and A := X/(M

N

). Splitting the range of q

into dyadic intervals and removing 1 − q

/Q by partial summation, we get

(2.1) |S|

 (M N )

Q

+ L

M

N Q

max

1≤Q1≤Q

|S(Q

)|, where

S(Q

) := X

q₁∼Q₁

X

n∼N

b

b

X

m∼M

m

e(Am

t).

If X(M N )

Q

≥ ε

, by Lemma 1.4 of [18] we transform the innermost sum to a sum over l and then by using Lemma 4 of [16, IV] with n = n we estimate the corresponding error term. As a result, we obtain

S(Q

)  X

q₁∼Q₁

X

n∼N

b

b

X

l∈I(n,q₁)

l

e(A

t

) + {(XM

N

Q

)

+ M

N Q

+ (X

M N Q

)

+ (X

M N

)

}L

,

where A

:=

A

l

, I(n, q

) := [c

AM

|t|, c

AM

|t|] and c

are some constants. Interchanging the order of summation and estimating the sum over l trivially, we find, for some l  X(M N )

Q

, the inequality

S(Q

)  (XM

N

Q

)

 X X

(n,q1)∈D(l)

b

b

e(A

t

)   (2.2) 

+ {(XM

N

Q

)

+ M

N Q

+ (X

M N Q

)

+ (X

M N

)

}L

, where D(l) is a subregion of {(n, q

) : n ∼ N, q

∼ Q

}. Let S

(Q

) be the double sums on the right-hand side of (2.2). Let Q

∈ (0, ε

min{Q

, N

/X}]

be another parameter to be chosen later. Using again the Cauchy–Schwarz inequality and Lemma 2.5 of [9] yields

(2.3) |S

(Q

)|

 (N Q

)

Q

+ N Q

Q

X

1≤q2≤Q2

|S

(q

, q

)|,

where

S

(q

, q

) := X

n∼N

X

q1∈J1(n)

b

b

e(t

(n, q

, q

)),

J

(n) is a subinterval of [Q

, 2Q

] and t

(n, q

, q

) := A

{t(n, q

+ q

)

− t(n, q

)

}. Putting n

:= n + q

, we have

S

(q

, q

)  X

n⁰∼N

 X

q₁∈J₂(n⁰)

e(t

(n

− q

, q

, q

))   ,

where J

(n

) is a subinterval of [Q

, 2Q

]. Noticing t(n

− q

, q

+ q

) − t(n

− q

, q

) = t(n

, q

),

t(n

− q

, q

+ q

) + t(n

− q

, q

) = 2t(n

− q

, q

) + t(n

, q

), we have

t

(n

− q

, q

, q

) = f (n

)q

+ r(n

, q

) + A

t(n

, q

)

,

where f (n

) := 2βA

t(n

, q

)n

and r(n

, q

) := 2A

t(n

−q

, q

)t(n

, q

)−

f (n

)q

. Since the last term on the right-hand side is independent of q

, it

follows that

S

(q

, q

)  X

n⁰∼N

 X

q1∈J2(n⁰)

e(±kf (n

)kq

+ r(n

, q

))   ,

where kak := min

|a − n|. Since Q

≤ ε

N

/X, we have

n

max

max

q1∈J2(n⁰)

|∂r/∂q

| ≤ c

XN

q

≤ 1/4.

By Lemmas 4.8, 4.2 and 4.4 of [21], the innermost sum on the right-hand side equals

\

J2(n⁰)

e(±kf (n

)ks + r(n

, s)) ds + O(1)

 kf (n

)k

if kf (n

)k ≥ ε

XN

q

, (XN

Q

q

)

if kf (n

)k < ε

XN

q

, which implies

S

(q

, q

)  X

kf (n⁰)k≥ε⁻¹₀ XN⁻²q₂

kf (n

)k

+ X

kf (n⁰)k<ε⁻¹₀ XN⁻²q₂

(XN

Q

q

)

=: S

+ S

.

As f

(n

)  XN

Q

q

, Lemma 3.1.2 of [14] yields S

 L

max

ε⁻¹₀ XN⁻²q2≤∆≤1/2

X

∆≤kf (n⁰)k<2∆

∆

 (N + X

N

Q

q

)L

,

S

 (XQ

q

)

+ (X

N

Q

q

)

. These imply, via (2.3),

|S

(Q

)|

 {(XN

Q

Q

)

+ (N Q

)

Q

+ (X

N

Q

Q

)

}L

, where we have used the fact that

N

Q

+ X

N

Q

Q

 (N Q

)

Q

(X ≥ N and Q

≥ Q

).

Using Lemma 2.4 of [9] to optimise Q

over (0, ε

min{Q

, N

/X}], we ob- tain

|S

(Q

)|

 {(XN

Q

)

+ (N

Q

)

+ N

Q

+ XQ

}L

,

where we have used the fact that (X

N

Q

)

and (N

Q

)

can be absorbed by (N

Q

)

(since X ≥ N ≥ Q

). Inserting this inequality into (2.2) yields

S(Q

)  {(X

M

N Q

)

+ (X

M

N Q

)

+ (XM

N Q

)

+ (X

M

N

Q

)

+ (X

M N Q

)

+ (X

M N

)

}L

,

where we have eliminated two superfluous terms (XM

N

Q

)

and M

N Q

. Replacing Q

by Q and inserting the estimate obtained into (2.1), we find

|S|

 {(X

M

N

Q

)

+ (X

M

N

Q

)

(2.4)

+ (X

M

N Q)

+ (M N )

Q

+ (XM

N

)

}L

,

where we have used the fact that (X

M

N

Q

)

and X

M

N

Q

can be absorbed by (M N )

Q

(since Q ≤ ε

N ≤ ε

X).

If X(M N )

Q

≤ ε

, we first remove m

by partial summation and then estimate the sum over m by the Kuz’min–Landau inequality ([9], The- orem 2.1). Therefore (2.4) always holds for 0 < Q ≤ ε

N . Optimising Q over (0, ε

N ] yields the desired result.

Next we consider a triple exponential sum S

:= X

m1∼M1

X

m2∼M2

X

m3∼M3

a

b

e



X m

m

m

M

M

M

 ,

which is a general form of S

. We have the following result.

Corollary 1. Let α ∈ R with α(α − 2) 6= 0, X > 0, M

≥ 1, |a

| ≤ 1,

|b

| ≤ 1 and let Y := 2 + XM

M

M

. Then

S

 {(X

M

M

M

)

+ (X

M

M

M

)

+ (X

M

M

M

)

+ (XM

M

M

)

+ (XM

)

M

+ M

(M

M

)

+ M

M

+ X

M

M

M

}Y

.

P r o o f. If M

:= X/M

≤ ε

, the Kuz’min–Landau inequality implies S

 X

M

M

M

. Next suppose M

≥ ε

. As before using Lemma 1.4 of [18] to the sum over m

and estimating the corresponding error term by Lemma 4 of [16, IV] with n = m

, we obtain

S

 X

M

S + (X

M

+ M

M

+ X

M

M

M

) log Y, where

S := X

m₁∼M₁

X

m₂∼M₂

X

m⁰₃∼M₃⁰

e a

eb

ξ

e



2X m

m

m

M

M

M



and |e a

| ≤ 1, |eb

| ≤ 1, |ξ

| ≤ 1. Let

M

:= M

M

and ξ e

:= X X

m₂m⁰₃=m⁰₂

eb

ξ

.

Then S can be written as a bilinear exponential sum S(M

, M

). Estimating it by Theorem 2 with (M, N ) = (M

, M

), we get the desired result.

Corollary 2. Let x

≤ M N ≤ ex

and |b

| ≤ 1. Then S

x

provided 1/2 ≤ θ < 1, H ≤ x

, M ≤ x

and N ≤ x

.

P r o o f. We apply Corollary 1 with (X, M

, M

, M

) = (xH/(M N ), N, H, M ).

3. Estimates for exponential sums S

. The main aim of this section is to prove the next Theorem 3. The inequality (3.1) improves Theorem 6 of [7] (or [18], Lemma 2) and the estimate (3.2) sharpens Lemma 14 of [1].

Theorem 3. Let α ∈ R with α 6= 0, 1, x > 0, H ≥ 1, M ≥ 1, N ≥ 1, X := xH/(M N ), |a

| ≤ 1 and |b

| ≤ 1. Let (κ, λ) be an exponent pair. If H ≤ N and HN ≤ X

, then

S

 {(X

H

M

N

)

+ (XH

M

N

)

+ (XH

M

N

)

(3.1)

+ (X

H

M

N

)

+ (H

M

N

)

+ (XH

M

N

)

+ (HM

N )

+ (X

HM

N

)

}x

,

S

 {(X

H

M

N

)

+ (HM

N )

(3.2)

+ (X

H

M

N

)

+ (X

HM

N

)

}x

.

The following corollary will be needed in the proof of Theorem 1.

Corollary 3. Let x

≤ M N ≤ ex

, |a

| ≤ 1 and |b

| ≤ 1. Then S

x

provided one of the following conditions holds:

1

2

≤ θ <

, H ≤ x

, x

≤ N ≤ x

; (3.3)

1

2

≤ θ <

, H ≤ x

, x

≤ N ≤ x

; (3.4)

1

2

≤ θ <

, H ≤ x

, x

≤ N ≤ x

; (3.5)

1

2

≤ θ <

, H ≤ x

, x

≤ N ≤ x

; (3.6)

1

2

≤ θ <

, H ≤ x

, x

≤ N ≤ x

; (3.7)

1

2

≤ θ <

, H ≤ x

, x

≤ N ≤ x

; (3.8)

1

2

≤ θ <

, H ≤ x

, x

≤ N ≤ x

. (3.9)

P r o o f. We obtain (3.3) from Lemma 9 of [1]. The result (3.4) is an

immediate consequence of (3.1). Let A and B be the classical A-process and

B-process. Taking, in (3.2),

(κ, λ) = BA

,

=

,

, (κ, λ) = BA

,

=

,

, (κ, λ) = BA

,

=

,

, (κ, λ) = BA

,

=

,

, (κ, λ) = BA

,

=

,

, we obtain (3.5)–(3.9). This completes the proof.

In order to prove Theorem 3, we need the next lemma. The first inequal- ity is essentially Theorem 2 of [7] with (M

, M

, M

, M

) = (H, M, N, 1), and the second one is a simple generalisation of Proposition 1 of [22]. It seems interesting that we prove (3.10) by an argument of Heath-Brown [11]

instead of the double large sieve inequality ([7], Proposition 1) as in [7].

Lemma 3.1. Let α, β ∈ R with αβ 6= 0, X > 0, H ≥ 1, M ≥ 1, N ≥ 1, L

:= log(2 + XHM N ), |a

| ≤ 1 and |b

| ≤ 1. Let f (h) ∈ C

[H, 2H]

satisfy the condition of exponent pair with f

(h)  F/H

(h ∼ H, k ∈ Z

) and

S = S(H, M, N ) := X

h∼H

X

m∼M

X

n∼N

a

b

e



X f (h)m

n

F M

N

 . If (κ, λ) is an exponent pair , then

S  {(XHM N )

+ H

M N + H(M N )

+ X

HM N }L

, (3.10)

S  {(X

H

M

N

)

+ H(M N )

+ H

M N (3.11)

+ X

HM N }L

.

P r o o f. Let Q ≥ 1 be a parameter to be chosen later and let M

:=

CM

N

where C is a suitable constant. Let T

:= {(m, n) : m ∼ M, n ∼ N, M

(q − 1) < m

n

Q ≤ M

q}. Then we can write

S = X

h∼H

a

X

q≤Q

X

(m,n)∈Tq

b

e



X f (h)m

n

F M

N

 .

By the Cauchy–Schwarz inequality, we have

|S|

 HQ X

q≤Q

X

(m,n)∈Tq

b

X

(fm,en)∈Tq

b

X

h∼H

e(g(h)) (3.12)

 HQ X

m,fm∼M

X

n,en∼N

|σ|≤M0/Q

   X

h∼H

e(g(h))  

 =: HQ(E

+ E

),

where σ := m

n

− e m

n e

, g(h) := Xσf (h)/(F M

N

) and E

, E

are the contributions corresponding to the cases |σ| ≤ M

/(M N ), M

/(M N ) < |σ|

≤ M

/Q, respectively.

Let D(M, N, ∆) := |{(m, e m, n, e n) : m, e m ∼ M ; n, e n ∼ N ; |σ| ≤ ∆M

}|.

By using Lemma 1 of [7], we find

(3.13) E

 HD(M, N, 1/(M N ))  HM N L

.

We prove (3.10) and (3.11) by using two different methods to esti- mate E

. Take Q := max{1, X/(ε

H)}. Then max

|g

(h)| = XH

∆ ≤ 1/2. The Kuz’min–Landau inequality implies

(3.14) E

 L

max

Q≤1/∆≤M N

D(M, N ; ∆)(XH

∆)

 X

H(M N )

L

. Now the inequality (3.10) follows from (3.12)–(3.14).

In view of (3.10), we can suppose X ≥ M N . Splitting (M

/(M N ), M

/Q]

into dyadic intervals (∆M

, 2∆M

] with Q ≤ 1/∆ ≤ M N and applying the exponent pair (κ, λ) yield

E

 L

max

Q≤1/∆≤M N

D(M, N ; ∆){(XH

∆)

H

+ (XH

∆)

} (3.15)

 (X

H

M

N

Q

+ X

HM

N

)L

.

Inserting (3.13) and (3.15) into (3.12) and noticing X

(HM N )

Q ≤ H

M N Q, we get

|S|

 {X

H

M

N

Q

+ H

M N Q}L

.

Using Lemma 2.4 of [9] to optimise Q over [1, ∞) yields the required re- sult (3.11).

Next we combine the methods of [1], [7] and [19] to prove Theorem 3.

Let Q

:= aH/(bN ) ∈ [100, HN ] be a parameter to be chosen later with a, b ∈ N and let Q

:= N Q

/( √

10H). Introducing T

:= {(h, n) : h ∼ H, n ∼ N, (q

− 1)/Q

≤ hn

< q

/Q

}, we may write

S

= X

q1≤Q1

X

m∼M

X X

(h,n)∈T_q1

a

b

e

 xh mn

 .

As before by the Cauchy–Schwarz inequality, we have (3.16) |S

|

M Q

X X

n1,n2∼N

X X

h1,h2∼H

|h₁/n₁−h₂/n₂|<1/Q^∗₁

b

b

δ

 h

n

, h

n

 X

m∼M

e

 x(h

n

− h

n

) mn

n

   ,

where δ(u

, u

) := |{q ∈ Z

: Q

max(u

, u

) < q ≤ Q

min(u

, u

) + 1}|.

Without loss of generality, we can suppose h

/n

≥ h

/n

in (3.16). Thus we have, with u

:= h

/n

,

δ(u

, u

) = [Q

u

+ 1] − [Q

u

] = 1 + Q

(u

− u

) − ψ(Q

u

) + ψ(Q

u

)

=: δ

+ δ

− δ

+ δ

,

where ψ(t) := {t} − 1/2 and {t} is the fractional part of t. Inserting into (3.16) yields

|S

|

 M Q

(|S

| + |S

| + |S

| + |S

|) with

S

:= X X

n₁,n₂∼N

X X

h1,h2∼H

|h1/n1−h2/n2|<1/Q^∗₁

b

b

δ

X

m∼M

e

 x(h

n

− h

n

) mn

n

 .

We estimate M Q

|S

| only; the other terms can be treated similarly. We write

M Q

|S

|  M Q

X X

n1,n2∼N

X

0≤kHN/Q₁

X X

h1,h2∼H h₁n₂−h₂n₁=k

δ

X

m∼M

e

 xk mn

n

   .

Since |δ

| ≤ 1, the terms with k = 0 contribute trivially O(HM

N Q

L

).

After dyadic split, we see that for some K with 1 ≤ K  HN/Q

and some D with 1 ≤ D ≤ min{K, N },

M Q

|S

|L

 M Q

X

d∼D

X X

n1,n2∼N⁰ (n1,n2)=1

X

r∼R

ω

(n

, n

; r) X

m∼M

e

 xr dmn

n

   

+ HM

N Q

, where N

:= N/D, R := K/D and

ω

(n

, n

; r) := X X

h1, h2∼H h1n2−h2n1=r

ψ(Q

h

/(dn

)).

In view of H ≤ N , Lemma 4 of [19] gives

|ω

(n

, n

; r)| =   

1

\

0

b

ω

(n

, n

; ϑ)e(rϑ) dϑ   (3.17) 

≤

1

\

0

|b ω

(n

, n

; ϑ)| dϑ  DL

, where

b

ω

(n

, n

; ϑ) := X

|m|≤8HN

ω

(n

, n

; m)e(−mϑ).

If L := XK/(HM N ) ≥ ε

, by Lemma 1.4 of [18] we transform the sum

over m into a sum over l, then we interchange the order of summations (r, l),

finally by Lemma 1.6 of [18] we relax the condition of summation of r. The

contribution of the main term of Lemma 1.4 of [18] is

(X

HM

N K

Q

)

× X

d∼D

X X

n1,n2∼N⁰ (n₁,n₂)=1

X

l∼L

   X

r∼R

g(r)e(rt)ω

(n

, n

; r)e(W p r/R)

  ,

where g(r) = (r/R)

, W := 2(XK/(HN ))(l/L)

(dn

n

/(DN

))

, t is a real number independent of variables. Let J := N

/D and τ

(j) :=

P

dn₁n₂=j

1. Let c

be some constants and

T

(j) := min{(X

HM

N

jr

)

, 1/kc

XH

M

N r/jk}.

By Lemma 4 of [16, IV], the contribution of the error term of Lemma 1.4 of [18] is

 DL

M Q

n

D

N

R+X

D

HM N

+ X

r∼R

X

j∼J

τ

(j)(T

(j) + T

(j)) o

 (HM N

+X

HM

N

Q

+X

HM N Q

+X

HM

N Q

)x

. Combining these and noticing X

HM

N Q

≤ HM

N Q

, we obtain

M Q

|S

|x

 (X

HM

N K

Q

)

S

+HM

N Q

(3.18)

+ X

HM

N

Q

+ X

HM N Q

+ HM N

, where

S

:= X

d∼D

X X

n₁,n₂∼N⁰ (n1,n2)=1

X

l∼L

   X

r∼R

g(r)e(rt)ω

(n

, n

; r)e(W p r/R)

  .

Let S

be the innermost sum. Using the Cauchy–Schwarz inequality and (3.17), we deduce

|S

|

 DL

1

\

0

|b ω

(n

, n

; ϑ)|

   X

r∼R

g(r)e(rt − rϑ)e(W p r/R)

dϑ.

By Lemma 2 of [7], we have, for any Q

∈ (0, R

],  

 X

r∼R

g(r)e(rt − rϑ)e(W p r/R)

≤ C



R

Q

+ RQ

X

1≤q₂≤Q₂

η X

r∼R

a

e

 W t(r, q

)

√ R



,

where C is a positive constant, η = η

= e

(1−|q

|/Q

), a

=

g(r + q

)g(r − q

), t(r, q

) := (r + q

)

− (r − q

)

. Splitting the range

of q

into dyadic intervals and inserting the preceding estimates into the

definition of S

, we find, for some Q

≤ Q

,

|S

|

 JL X

d∼D

X X

n1,n2∼N⁰ (n1,n2)=1

X

l∼L

|S

|

(3.19)

 D

L

{(JLR)

Q

+ JLRQ

S

}, where Z := 2XK/(HN ) and

S

:= X

q₂∼Q_2,0

X

j∼J

τ

(j)    

X

l∼L

X

r∼R

a

e



Z (l/j)

t(r, q

) (LR/J)

   .

Applying (3.10) of Lemma 3.1 with (X, H, M, N ) = (ZR

q

, R, J, L) to the inner triple sums and summing trivially over q

, we find

S

 {(ZJLQ

)

+ (JL)

RQ

+ JLR

Q

+ (Z

J

L

R

Q

)

}x

.

Replacing Q

by Q

and inserting the estimate obtained into (3.19) yield S

 {(ZJ

L

R

Q

)

+ JLRQ

+ (Z

J

L

R

Q

)

+ (JL)

R + JLR

}Dx

.

Using Lemma 2.4 of [9] to optimise Q

over (0, R

], we find

|S

|  {(ZJ

L

R

)

+ (JL)

R + JLR

}Dx

,

where for simplifying we have used the fact that JLR

≤ JLR

, (JLR)

= {(JL)

R}

{JLR

}

, Z

JLR ≤ JLR

. Inserting J = D

N

, L = XK/(HM N ), R = D

K, Z = 2XK/(HN ), we ob- tain an estimate for S

in terms of (X, D, H, M, N, K). Noticing that all exponents of D are negative, we can replace D by 1 to write

|S

|  {(X

H

M

N

K

)

+ (X

H

M

N

K

)

+ (X

H

M

N

K

)

}x

.

Inserting into (3.18) and replacing K by HN/Q

yield

M Q

|S

|  {(X

H

M

N

Q

)

+ (XH

M

N

Q

)

(3.20)

+ (X

H

M

N

Q

)

+ HM

N Q

+ X

HM

N

Q

}x

=: E(Q

)x

, where we have used the fact that

X

HM N Q

+ HM N

 (X

H

M

N

Q

)

.

If L ≤ ε

, using the Kuz’min–Landau inequality and (3.17) yields M Q

|S

|L

 M Q

D

N

RDL

/L  X

HM

N

Q

L

 E(Q

)L

. Therefore the estimate (3.20) always holds. Similarly we can establish the same bound for M Q

|S

| (j = 1, 2, 4). Hence we obtain, for any Q

∈ [100, HN ],

|S

|

 E(Q

)x

.

In view of the term HM

N Q

, this inequality is trivial when Q

≥ HN . By using Lemma 2.4 of [9], we see that there exists some e Q

∈ [100, ∞) such that

E( e Q

)  (X

H

M

N

)

+ (XH

M

N

)

+ (X

H

M

N

)

+ (X

H

M

N

)

+ (H

M

N

)

+ (XH

M

N

)

+ HM

N + X

HM

N

.

Now taking Q

:= 100[ e Q

]H(1 + [N ])/((1 + [H])N ) and noticing that E(Q

)  E( e Q

), we obtain the desired result (3.1).

In order to prove (3.2), we first write S

= X

d∼D

X X

n1,n2∼N⁰ (n1,n2)=1

X

l∼L

1

\

0

b

ω

(n

, n

; ϑ)S

(ϑ) dϑ   ,

where S

(ϑ) = P

r∼R

g(r)e(f (r)), f (r) = W p

r/R + (t + ϑ)r (t, ϑ ∈ [0, 1]). Since HN ≤ X

, we have

f

(r)  W/R + t + ϑ  LM/R + t + ϑ ≥ LM/K + t + ϑ ≥ (HN )

. Removing the smooth coefficient g(r) by partial summation and using the exponent pair (κ, λ) yield the inequality S

(ϑ)  (W/R)

R

uni- formly for ϑ ∈ [0, 1]. Thus by (3.17), we find

S

 JL(W/R)

R

DL

 X

H

M

N

K

L

, which implies, via (3.18),

M Q

|S

|  (X

H

M N

Q

+ HM

N Q

+ X

HM

N

Q

)x

, where we have used the fact that

X

HM N Q

+ HM N

 X

H

M N

Q

. The same estimate holds also for M Q

|S

| (j = 1, 2, 4). Thus we obtain, for any Q

∈ [100, HN ],

|S

|

 (X

H

M N

Q

+HM

N Q

+X

HM

N

Q

)x

.

This implies (3.2). The proof of Theorem 3 is finished.