1. Introduction. Let λ

LXXXI.3 (1997)

Diagonal cubic equations

by

Hongze Li (Jinan)

1. Introduction. Let λ

be positive integers. We shall be concerned with solutions of the diophantine equation

(1.1) λ

x

+ λ

x

+ . . . + λ

x

= 0

which are small in the following sense. Let Λ = λ

λ

. . . λ

. Then we seek for solutions to (1.1) which satisfy

(1.2) 0 <

X

λ

|x

|

 Λ

with g as small as possible.

A first positive answer to the problem was found by R. C. Baker [1]. He established the solubility of (1.1) and (1.2) whenever g = 61. In this paper we prove the following theorem.

Theorem. Let g = 14. Then there are solutions to (1.1), (1.2).

The key for improvement is the mean value estimate for a class of ex- ponential sums presented in [2]. We state the special case we require as a lemma. Let X be a large parameter, and write

(1.3) θ

= 15/113, θ

= 14/113, η = 84/113.

We note that θ

+ θ

+ η = 1. Put

(1.4) M

= X

, M

= X

, Y = X

.

As usual we write e(α) = exp(2πiα). Now define the exponential sums

(1.5) g(α, P ) = X

P <X≤2P

e(αx

),

1991 Mathematics Subject Classification: Primary 11P55.

This work was supported by the National Natural Science Foundation of China.

[199]

(1.6) G(α, X) = X

M₁<p₁≤2M₁

X

M₂<p₂≤2M₂

g(α(p

p

)

, Y ), where, here and throughout, p, p

denote prime numbers.

Lemma 1. In the notation introduced above, for any ε > 0 we have

1

\

0

|G(α, X)|

dα  X

. This is the case l = 3 of the Theorem in Br¨ udern [2].

2. Preliminaries. Now define (2.1) χ

(X)

= {p

p

y : X

< p

≤ 2X

, X

< p

≤ 2X

, X

< y ≤ 2X

} and

(2.2) χ

(X) = {py : X

< p ≤ 2X

, y ∈ χ

(X

)}

in which θ = 233/1815.

Given positive integers µ

, µ

, we write S(µ

, µ

, B, E) for the number of solutions of the equation

(2.3) µ

x

+ µ

p

(y

+ y

) = µ

x

+ µ

p

(y

+ y

) in integers x

, x

, y

, y

, y

, y

and primes p satisfying

B < x

, x

≤ 2B, p - µ

x

x

, y

∈ χ

(E

) (1 ≤ i ≤ 4), (2.4)

E

< p ≤ 2E

, p ≡ 2 (mod 3).

(2.5)

Lemma 2. Let B ≥ 1, E ≥ 1. Let µ

, µ

be positive integers, and (2.6) µ

E

 µ

B

 µ

E

, µ

 µ

, E

≤ B

. Then

(2.7) S(µ

, µ

, B, E)  B

E

.

P r o o f. By the argument of Vaughan [7], p. 125, we have S(µ

, µ

, B, E)

 E

S

, where S

is the number of solutions to (2.3) subject to (2.4), (2.5) and x

≡ x

(mod p

). The solutions with x

= x

contribute to S

at most BE

E

= BE

. For the solutions with x

6= x

we again follow through the argument in Vaughan [7] and find that there are at most 2S

remaining solutions where S

equals the number of solutions to

µ

h(3x

+ h

p

) = 4µ

(y

+ y

− y

− y

),

where x ≤ 4B, E

< p ≤ 2E

, 0 < h ≤ H, y

∈ χ

(E

), and H = CBE

for some constant C. On writing

Ω(α) = X

E^θ<p≤2E^θ

X

h≤H

X

x≤4B

e(αh(3x

+ h

p

))

we have, by H¨older’s inequality, S

=

1

\

0

Ω(µ

α)|G(4µ

α, E

)|

dα

≤



\

0

|Ω(µ

α)|

dα





\

0

|G(α, E

)|

dα



.

The argument of Vaughan [8], p. 39 shows that for B

< E

≤ B

we have

1

\

0

|Ω(α)|

dα  H

(E

B)

. Since Ω(α) has period 1, we have

1

\

0

|Ω(µ

α)|

dα = 1 µ

µ

\

0

|Ω(α)|

dα =

1

\

0

|Ω(α)|

dα and by Lemma 1 we have

1

\

0

|G(α, E

)|

dα  (E

)

. The lemma follows easily.

3. Outline of the method. For the proof of the Theorem, as in Baker [1] it suffices to prove the Theorem in the special case when λ

, . . . , λ

are cube-free, and no prime divides more than four of the λ

. We may suppose that

(3.1) λ

≥ λ

≥ λ

≥ λ

≥ λ

≥ λ

≥ λ

. Let

(3.2) N = C

Λ

, P = N

, P

= P λ

,

where C

is sufficiently large. Let δ denote a sufficiently small positive ab- solute constant, and let ε = δ

.

Let R(λ

, . . . , λ

) denote the number of solutions of the equation (3.3) λ

x

− λ

p

y

+ λ

p

p

p

p

z

+ λ

p

p

p

p

z

+ λ

u

+ λ

p

p

p

v

+ λ

p

p

p

v

= 0

in integers x, y, z

, z

, u, v

, v

and primes p, p

satisfying (3.4) P

< x ≤ 2P

, W < y ≤ 2W, R

< z

≤ 2R

,

U < u ≤ 2U, V

< v

≤ 2V

,

(3.5) p - x, p

- y (3 ≤ i ≤ 4, 1 ≤ j ≤ 3), p

- u (6 ≤ i ≤ 7, 1 ≤ j ≤ 3), (3.6) Y < p ≤ 2Y, p ≡ 2 (mod 3), p - Λ,

(3.7) Z

< p

≤ 2Z

, p

≡ 2 (mod 3), (i = 3, 4, 6, 7, 1 ≤ j ≤ 3), p

- Λ.

Here

(3.8) Y = C

P

, W = P

Y

, R

= 1

200 (P

Y

)

(i = 3, 4),

(3.9) U = 1

200 P

, V

= 1

200 P

(i = 6, 7), (3.10) Z

= 2

(P

Y

)

, Z

= 2

(P

Y

)

,

Z

= 2

(P

Y

)

(i = 3, 4), (3.11) Z

= 2

P

, Z

= 2

P

,

Z

= 2

P

(i = 6, 7), where C

is sufficiently large.

The inequalities

(3.12) N  λ

P

, λ

Y

W

, λ

Y

Z

Z

Z

R

, λ

Y

Z

Z

Z

R

, λ

U

, λ

Z

Z

Z

V

, λ

Z

Z

Z

V

 N are easily verified.

Let (a

, a

) or more generally (a

, . . . , a

) denote greatest common divi- sor, while [a

, . . . , a

] denotes least common multiple. We write

(3.13) f

(X, α) = X

X<x≤2X (x,d)=1

e(αx

)

and use the notation f in place of f

.

Let p denote the ordered set p, p

, p

, p

, p

, p

, p

, p

, p

, p

, p

, p

, p

and let

(3.14) A =

Y

Y

p

, B = Y

Y

p

,

(3.15) F (p, α) = f

(P

, λ

α)f

(W, λ

p

α)f (R

, λ

p

p

p

p

α)

× f (R

, λ

p

p

p

p

α)

× f

(U, λ

α)f (V

, λ

p

p

p

α)f (V

, λ

p

p

p

α),

(3.16) F (α) = X

p (3.6),(3.7)

F (p; α).

The summation here is over ordered sets p satisfying (3.6), (3.7); we often use this type of notation below.

Let < be the unit interval [LN

, 1 + LN

]. Then clearly (3.17) R(λ

, . . . , λ

) = \

<

F (α) dα.

Here

(3.18) L = P

/10.

When 1 ≤ a ≤ q ≤ L and (a, q) = 1, we take M (q, a) to be the interval {α : |α − a/q| ≤ q

LN

}, and let M denote the union of all such M (q, a).

Then M (q, a) are disjoint subsets of M , as we easily verify using (3.2) and (3.18). Let

(3.19) m = < \ M.

We shall prove that

\

M

F (α) dα  λ

Λ

N

(logN )

, (3.20)

\

m

F (α) dα  λ

Λ

N

. (3.21)

It follows from (3.19)–(3.21) and (3.17) that

(3.22) R(λ

, . . . , λ

)  λ

Λ

N

(log N )

. Thus the Theorem will follow once (3.20)–(3.21) have been proved.

We recall the standard notations (3.23) S(q, b) =

X

e(br

/q), J(β, X) =

2X

\

X

e(βx

) dx.

The estimate

(3.24) S(q, b)  q

ψ(q), (q, b) = 1, where ψ(q) is the multiplicative function with

(3.25) ψ(p

) = p

(h = 0, 1, . . . ; 0 ≤ r ≤ 2), follows from Lemmas 4.3 and 4.4 of [5]. By partial integration,

(3.26) J(β, X)  X

1 + X

|β|

for positive X and real β.

Lemma 3. Let d be an integer with  1 divisors. Let (3.27) s

(q, c) = q

X

b|d

µ(d) S(q, b

c) b and let λ be a positive integer. Then

f

(X, λα) = s

(q, λa)J(λ(α − a/q), X) (3.28)

+ O(q

(1 + λX

|α − a/q|)

) for any X > 0, real α and rational number a/q.

This is Lemma 2 of [1].

For α ∈ M (q, a), we introduce the notations s(p, α) = s

(q, λ

a)J(λ

(α − a/q), P

), (3.29)

g(p, α) = s

(q, λ

p

a)J(λ

(α − a/q)p

, W ).

(3.30)

It is convenient to write

(3.31) ∆ = ∆(α, a, q) = 1 + N |α − a/q|.

Lemma 4. Let p satisfy (3.6) and (3.7). Let α ∈ M (q, a) where 1 ≤ a ≤ q ≤ L, (a, q) = 1. Then

f

(P

, λ

α) − s(p, α)  q

∆

, (3.32)

f

(W, λ

p

α) − g(p, α)  q

∆

. (3.33)

P r o o f. The proof is similar to that of Lemma 3 of [1]; see [1] and [6].

4. The minor arcs. In this section we use the notations (4.1) H = C

P

Y

, Q = P

Y

,

where C

is a sufficiently large absolute constant; also

(4.2) M = L(2Y )

.

Further, let (4.3) S(α) =

 X

p31,p32,p33

(3.7)

X

p41,p42,p43

(3.7)

f

(W, λ

α)

× f (R

, λ

p

p

p

α)f (R

, λ

p

p

p

α)

and

Φ

(α) = X

P₁<y≤2P₁ p|y

1 (4.4)

+ 2Re X

h≤H

X

2P1+hp³<y≤4P1−hp³ p-y, y≡h (mod 2)

e

 λ

α

4 (3hy

+ h

p

)

 .

Let n denote the set of real numbers in (0, 1] with the property that whenever |α − a/q| ≤ q

LN

and (a, q) = 1, we have q > M . Let

(4.5) T = \

n

X

p (3.6)

Φ

(α)S(α) dα.

Lemma 5. Let J

=

1

\

0

 X

p61,p62,p63

(3.7)

X

p71,p72,p73

(3.7)

f

(U, λ

α)

× f (V

, λ

p

p

p

α)f (V

, λ

p

p

p

α)   

dα.

Then J

 P

P

P

.

P r o o f. By considering the underlying diophantine equation we have J

1

\

0

 X

p61,p62,p63

(3.7)

X

p71,p72,p73

(3.7)

f

(U, λ

α)f (V

, λ

p

p

p

α)

× f (V

, λ

p

p

p

α)   

dα.

From the inequality |zw| ≤

|z|

+

|w|

, we have J

X

\

0

 X

p61

(3.7)

X

p71

(3.7)

|f

(U, λ

α)|

 X

pk2,pk3

(3.7)

f (V

, λ

p

p

p

α)   



dα.

Hence, by Cauchy’s inequality, J

 Z

Z

X

\

0

X

p61

(3.7)

X

p71

(3.7)

|f

(U, λ

α)|

×  

 X

p_k2,p_k3 (3.7)

f (V

, λ

p

p

p

α)   

dα

 Z

Z

S(λ

, λ

, P

, P

) + Z

Z

S(λ

, λ

, P

, P

).

By (3.1) and (3.2) we have

P

≤ P

, P

≤ P

≤ P

. By (3.11), (3.12) and Lemma 2,

J

 P

P

P

. This completes the proof of Lemma 5.

Lemma 6. Let J

= \

m

   X

p (3.6)

X

p31,p32,p33

(3.7)

X

p41,p42,p43

(3.7)

f

(P

, λ

α)f

(W, λ

p

α)

× f (R

, λ

p

p

p

p

α)f (R

, λ

p

p

p

p

α)   

dα.

Then J

 Y T.

P r o o f. The proof is similar to that of Lemma 15 of [1].

Let

F (β, γ; h) = X

2P1<y≤4P1

y≡h (mod 2)

e

βy

− γy
, (4.6)

G

(%, σ) = X

Y <p≤2Y, p≡2 (mod 3) p≤(2P1/h)^1/3

e

%p

+ σp

, (4.7)

F

(α; h) = X

2P1+hp³<y≤4P1−hp³ y≡h (mod 2)

e

λ

αhy

, (4.8)

Ψ

(α) = 2Re X

h≤H

F

(α; h)e

λ

αh

p

, (4.9)

D

(α; h) = X

2P1/p+hp²<y≤4P1/p−hp² y≡h (mod 2)

e

λ

αhp

y

, (4.10)

Ξ

(α) = 2Re X

h≤H

D

(α; h)e

λ

αh

p

, (4.11)

T

(p) = \

n

Ψ

(α)S(α) dα, (4.12)

T

(p) = \

n

Ξ

(α)S(α) dα, (4.13)

T

=

1

\

0

S(α) dα.

(4.14)

By (4.4),

(4.15) Φ

(α) = Ψ

(α) − Ξ

(α) + O(P

).

Hence

(4.16) T = X

p (3.6)

(T

(p) − T

(p)) + O(P

Y T

)

from (4.5).

We estimate T

via Lemma 2. By (3.1), (3.2), (3.8) and (3.10) we have (P

Y

)

≤ (P

Y

)

≤ (P

Y

)

.

Just as in Lemma 5, we have

(4.17) T

 (P

Y

)(P

Y

)

(P

Y

)

 P

P

P

Y

. Let

(4.18) T

(γ, θ) = \

n

X

h≤H

|F (αλ

h, γ; h)G

(αλ

h

, θγh)|S(α) dα.

By a very minor adaptation of the proof of (5.26) of [7], one finds that

(4.19) X

p (3.6)

T

(p)  (log P

) sup

0≤γ≤1 θ=±1

T

(γ, θ) + N

Λ

.

We omit the details.

Lemma 7. Suppose that α ∈ R and  

 α − a q  

  ≤ 1

24qHP

, (a, q) = 1.

Then

X

h≤H

|F (αh, γ; h)|

 P

 q

HP

1 + Q

|α − a/q| + HP

+ q

 . This is Lemma 16 of [1].

Lemma 8. Let α, γ ∈ R. Suppose that  

 α − a q  

  ≤ q

Q

H

, (a, q) = 1, q ≤ Q

H

. Then

X

h≤H

|G

(αh

, γh)|

 P

 HY

q

(1 + Q

|α − a/q|)

+ H

Y



.

This is essentially Lemma 8 of [7].

Lemma 9. Let γ ∈ R and θ ∈ {−1, 1}. Then

T

(γ, θ)  P

P

P

P

Y

+ λ

P

P

P

Y

+ λ

P

P

P

P

Y

.

P r o o f. Let n

denote the set of α in n with the property that whenever (4.20) |λ

α − r/b| ≤ r

H

Q

, (b, r) = 1,

one has r > H

. Let α ∈ n

. We apply Lemma 7 with λ

α in place of α. By Dirichlet’s theorem there exist integers b, r satisfying (4.20) with r ≤ Q

H

; here we must have r > H

. Now the condition of Lemma 7 is easily verified, thus

X

h≤H

|F (αhλ

, γ; h)|

 P

(P

H

+ HP

+ Q

H

) (4.21)

 P

H

.

We may choose c, s so that (c, s) = 1, s ≤ Q

H

, and

|λ

α − c/s| ≤ s

H

Q

.

Then |λ

α − c/s| ≤ s

H

Q

, so that s > H

> H

. By Lemma 8, applied to λ

α in place of α, we have

X

h≤H

|G

(λ

αh

, θγh)|

 P

H

Y

. Hence by Cauchy’s inequality, (4.14), (4.17) and (4.21), (4.22) \

n₁

X

h≤H

|F (αhλ

, γ; h)G

(λ

αh

, θγh)|S(α) dα

 P

Y T

 P

P

P

P

Y

. It remains to consider n \ n

. Let α ∈ n \ n

. There are integers b, r satisfying

(4.23) |λ

α − b/r| ≤ r

H

Q

, (b, r) = 1, r ≤ H

.

We write b/(λ

r) = a/q in lowest terms. Clearly q = rd where d = λ

/(λ

, b).

Also, (r, λ

/d) = (r, (λ

, b)) = 1. Because α ∈ n, we know that either

|α − a/(rd)| > r

d

LN

, or rd > M , or both.

Let d be a given divisor of λ

. Let N

(d, r, a) =

 α :

   α − a

rd  

  ≤ r

d

LN

 , N

(d, r, a) =

 α :

   α − a

rd  

  ≤ r

H

Q

λ

 .

It is clear from (3.2), (3.8), (3.18), (4.1) that N

(d, r, a) contains N

(d, r, a).

Let N

(d) denote the union of N

(d, r, a) with 1 ≤ a ≤ rd, (a, rd) = (r, λ

/d) = 1, (4.24)

r ≤ M d

. (4.25)

Let N

(d) denote the union of N

(d, r, a) with (4.24) and r ≤ H

. By the discussion following (4.22) we have, modulo one,

n \ n

⊂ [

d|λ₁

{N

(d) \ N

(d)}.

Let L(d, r, a) denote N

(d, r, a) when M d

< r ≤ H

, and (4.24) holds;

and denote N

(d, r, a) \ N

(d, r, a) when r ≤ M d

and (4.24) holds. For fixed d, we have

N

(d) \ N

(d) = [

r≤H^7/4;a

L(d, r, a).

Let α ∈ L(d, r, a). The fraction λ

a/(rd) may be written in the form b/r with (b, r) = 1. Since

(λ

a, rd) = d(λ

a/d, r) = d, by Lemma 7, with λ

α in place of α, and

 λ

α − b r     = λ

   α − a

rd  

  ≤ r

H

Q

≤ 1 24rHP

, we have

X

h≤H

|F (αhλ

, γ; h)|

 P

 r

HP

1 + Q

|λ

α − b/r| + HP

+ r

 (4.26)

 r

HP

1 + N Y

|α − a/(rd)| . Here we require the observations that

Q

λ

= N Y

, HP

+ r  HP

 r

HP

, (HP

+ r)rQ

|λ

α − b/r|  HP

H

 HP

.

Choose c, s so that |λ

α − c/s| ≤ s

H

Q

, (c, s) = 1 and s ≤ Q

H

. If b/r = c/s, then by the definition of L(d, r, a), we know that either

|λ

α − c/s| > s

d

LN

λ

≥ s

LN

,

or s > M d

, or both. By (3.1), (3.2) we have either s > H

or Q

s|λ

α − c/s| ≥ H

, or both. If b/r 6= c/s, then

1 rs ≤

    b

r − c s  

  ≤ (H

s

+ H

r

)Q

≤ 1

2sr + H

r

Q

,

whence s >

Q

H

> H

once more. Now we may apply Lemma 8 with λ

α in place of α, obtaining

(4.27) X

h≤H

|G

(λ

αh

, θγh)|

 P

H

Y

.

Therefore, by (4.26) and Cauchy’s inequality (4.28) X

h≤H

|F (αhλ

, γ; h)G

(λ

αh

, θγh)|  Y H

P

r

(1 + N Y

|α − a/(rd)|)

. It follows that

(4.29) \

N2(d)\N1(d)

X

h≤H

|F (αhλ

, γ; h)G

(λ

αh

, θγh)|S(α) dα

 P

H

Y \

N2(d)\N1(d)

r



1 + N Y

   α − a

rd    



S(α) dα.

We write

Γ = Γ (α, a, q) = q

(1 + N Y

|α − a/q|)

. By Lemma 3, we have

f

(W, λ

α) = s

(q, λ

a)J(λ

(α − a/q), W ) + O(Γ ) = Γ

+ O(Γ ).

Hence by (3.26) we have

|f

(W, λ

α)|

 Γ

+ Γ

 |s

(q, λ

a)J(λ

(α − a/q), W )|

+ Γ

 s

(q, λ

a)W

(1 + N Y

|α − a/q|)

+ Γ

. Hence

(4.30) \

N2(d)\N1(d)

r



1 + N Y

   α − a

rd    



S(α) dα

 \

N₂(d)\N₁(d)

r



1 + N Y

   α − a

rd    



(Γ

+ Γ

)

×  

 X

p₃₁,p₃₂,p₃₃ (3.7)

X

p₄₁,p₄₂,p₄₃ (3.7)

f (R

, λ

p

p

p

α)f (R

, λ

p

p

p

α)   

dα.

By Hua’s inequality we have

1

\

0

 X

p31,p32,p33

(3.7)

f (R

, λ

p

p

p

α)  

dα  (P

Y

)

,

1

\

0

 X

p41,p42,p43

(3.7)

f (R

, λ

p

p

p

α)  

dα  (P

Y

)

and

1

\

0

 X

p31,p32,p33

(3.7)

X

p41,p42,p43

(3.7)

f (R

, λ

p

p

p

α)f (R

, λ

p

p

p

α)   

dα



\

0

 X

p31,p32,p33

(3.7)

f (R

, λ

p

p

p

α)   

dα



×



\

0

 X

p41,p42,p43

(3.7)

f (R

, λ

p

p

p

α)   

dα



 ((P

Y

)

(P

Y

)

)

. By the definition of Γ ,

Γ

= q



1 + N Y

   α − a

q    



= (rd)



1 + N Y

   α − a

rd    

 . For α ∈ N

(d) \ N

(d),

r

d



1 + N Y

   α − a

rd    



 d

H

 λ

H

. Hence

(4.31) \

N₂(d)\N₁(d)

r



1 + N Y

   α − a

rd    



Γ

× | X

p₃₁,p₃₂,p₃₃ (3.7)

X

p₄₁,p₄₂,p₄₃ (3.7)

f (R

, λ

p

p

p

α)f (R

, λ

p

p

p

α)|

dα

 \

N2(d)\N1(d)

r

d



1 + N Y

   α − a

rd    



×  

 X

p₃₁,p₃₂,p₃₃ (3.7)

X

p₄₁,p₄₂,p₄₃ (3.7)

f (R

, λ

p

p

p

α)f (R

, λ

p

p

p

α)   

dα

 λ

H

(P

P

Y

)

.

Now we consider

(4.32) \

N₂(d)\N₁(d)

Γ

dα = \

N₂(d)\N₁(d)

s

(q, λ

a)W

(1 + N Y

|α − a/(rd)|)

dα

 W

X

r≤H^7/4

X

a≤rd

\

L(d,r,a)

s

(q, λ

a)

(1 + N Y

|α − a/(rd)|)

dα.

Let κ(q) and κ

(q) be the multiplicative functions defined by κ(p

) = p

, κ(p

) = 2p

, κ(p

) = p

, κ

(p) = 2p

, κ

(p

) = p

, κ

(p

) = p

(l ≥ 3).

Then κ(q) ≤ κ

(q), and by Lemmas 4.3, 4.4 and Theorem 4.2 of Vaughan [5], q

S(q, a)  κ(q) when (a, q) = 1.

Now we consider X

q≤Ξ

qκ

 q

(q, λ

)



 Y

p≤Ξ

 1 +

X

p

κ

 p

(p

, λ

)





 Y

p-λ₂ p≤Ξ



1 + 16p

+ p

+ X

p



× Y

pkλ2

p≤Ξ



1 + p + 16 + 1 + 1 + X

l=1

p



× Y

p²kλ₂ p≤Ξ



1 + p + p

+ 16p + p + p + p

+ p

+ X

l=0

p

 .

We have Y

pkλ₂ p≤Ξ

(20 + p)  Y

pkλ₂ p≤C(ε)

(20 + p) Y

pkλ₂ p≥C(ε)

(20 + p)  Y

pkλ₂

p

.

Here we use the fact that for p ≥ C(ε) we have 20 + p ≤ p

. Similarly Y

p²kλ2

p≤Ξ

(p

+ 22p + 1)  Y

p²kλ2

p

.

The constant implied by  depends only on ε. So we have

(4.33) X

q≤Ξ

qκ

 q

(q, λ

)



 λ

(log Ξ)

. By (3.9) of [1],

s

(q, λ

a)  q

κ

 q

(q, λ

a)



 q

κ

 q

(q, λ

)

 , so we have

(4.34) \

N2(d)\N1(d)

Γ

dα  W

Y

N

λ

.

For α ∈ N

(d) \ N

(d), we have r



1 + N Y

   α − a

rd    



 M

d

. Hence by Schwarz’s inequality,

(4.35) \

N2(d)\N1(d)

r



1 + N Y

   α − a

rd    



Γ

×  

 X

p31,p32,p33

(3.7)

X

p41,p42,p43

(3.7)

f (R

, λ

p

p

p

α)f (R

, λ

p

p

p

α)   

dα

 (W

Y

N

λ

)

M

d

(P

Y

)

(P

Y

)

. By (4.29)–(4.35) we have

(4.36) \

n\n1

X

h≤H

|F (αhλ

, γ; h)G

(λ

αh

, θγh)|S(α) dα

 P

H

Y [λ

H

(P

P

Y

)

+ (W

Y

N

λ

)

M

d

(P

P

Y

)

]

 λ

P

P

P

Y

+ λ

P

P

P

P

Y

. By (4.22), (4.36) the lemma follows.

As in Section 8 of [1], for the estimation of P

p,(3.6)

T

(p) we have the same upper bound for T

(γ, θ).

5. First steps on the major arcs. In this section we show that R(λ

, . . . , λ

) is well approximated by integrals similar to (3.17), but with f

(P

, λ

α) and f

(W, λ

p

α) replaced by suitable approximations. Let

D(α) = q

(1 + N |α − a/q|)

.

By Lemma 4,

f

(P

, λ

α) − s(p, α)  q

∆

 D(α)q

, f

(W, λ

α) − g(p, α)  q

∆

 D(α)q

. Here

∆ = 1 + N |α − a/q|.

Now we introduce the numbers v

= \

M

   X

p (3.6)

X

p₃₁,p₃₂,p₃₃ (3.7)

X

p₄₁,p₄₂,p₄₃ (3.7)

(f

(P

, λ

α) − s(p, α))f

(W, λ

p

α)

× f (R

, λ

p

p

p

p

α)f (R

, λ

p

p

p

p

α)   

dα, v

= \

M

   X

p (3.6)

X

p31,p32,p33

(3.7)

X

p41,p42,p43

(3.7)

s(p, α)(f

(W, λ

α) − g(p, α))

× f (R

, λ

p

p

p

p

α)f (R

, λ

p

p

p

p

α)   

dα.

We use Cauchy’s inequality and note that D(α)  L

for α ∈ M . Hence by (4.17),

v

 Y L

X

Y <p≤2Y 1

\

0

 X

p31,p32,p33

(3.7)

X

p41,p42,p43

(3.7)

f

(W, λ

p

α) (5.1)

× f (R

, λ

p

p

p

p

α)f (R

, λ

p

p

p

p

α)   

dα

 Y

L

1

\

0

 X

p₃₁,p₃₂,p₃₃ (3.7)

X

p₄₁,p₄₂,p₄₃ (3.7)

f

(W, λ

α)

× f (R

, λ

p

p

p

α)f (R

, λ

p

p

p

α)   

dα

 Y

L

T

 L

P

P

P

Y

.

Now we consider v

. By Cauchy’s inequality and Schwarz’s inequality we have

v

 Y P

X

Y <p≤2Y

\

M

s(p, α)D(α)

× X

p₃₁,p₃₂,p₃₃ (3.7)

X

p₄₁,p₄₂,p₄₃ (3.7)

f (R

, λ

p

p

p

p

α)f (R

, λ

p

p

p

p

α)   

dα

 Y P

(I

I

)

,