Mean-value estimates for nonlinear Weyl sums over primes

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Mean-value

estimates

for

nonlinear

Weyl

sums

over

primes

By

Jianya Liu and Jingmei YE

(Received November 25, 2003)

(Revised September 21, 2004)

(from Nagoya Mathematical Journal)

Abstract.

We establish a Bombieri-type mean-value estimate for nonlinear ex

ponential sums over primes. This has applications in the Waring-Goldbach problem.

1. Introduction

and statement

of results

Estimates

for exponential

sums over primes play important

roles in the addi

tive theory of prime numbers.

In this note we are concerned with the sums

where k is a positive integer, •È(n) the von Mangoldt function, and e(a)=e2ƒÎia. Here and throughout, m •` y means y/2<m_??_y. Let ƒ¿ •¸ [0,1] satisfy the rational

approximation

(1.1)

For small q and ƒÉ compared with y, for example when q_??_logAy with A>0

arbitrary and •bƒÉ•b< y-k exp(c1 logy), we have the asymptotic formula (see e.g. [7], Chap. 6)

(1.2)

where c1, c2 are positive constants, _??_(n)

the Euler totient function, and

(1.3)

2000 Mathematics Subject Classification. 11L07, 11L20.

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380 JIANYA LIU and JINGMEI YE

For big q, we have the following classical bound of Vinogradov: For any A>0 there exists B=B(A)>0, such that if q_??_logB y and •bƒÉ•b_??_q-2, then Sk (y, a)<<

ylog-A y

In the case k=1, Wolke [8] studied the mean-value of the error term in (1.2),

and proved the following theorem of Bombieri-Vinogradov type. Note that when

k=1 and (a, q)=1, we have C1(q, a)=ƒÊ(q), the Mobius function. Actually Wolke

proved that for any A>0, there exists a constant B=B(A)>0, such that

(1.4)

if

where here and throughout L stands for log x. Later Zhan and the first author [5]

improved the above result to the extent that (1.4) is true for the bigger range

(1.5)

This result coincides with a consequence of the Generalized Riemann Hypothesis,

and has been applied to the additive theory of prime numbers; see e.g. [8], [5].

In this note, we extend our theorem to general k.

THEOREM 1.1. Fix k_??_2. Let a be as in (1.1) and ƒÃ>0 arbitrary. For any A>0, there exists a constant B=B(A)>0, such that if Q and ƒÆ satisfy

(1.6)

then

The factor q-1/2 on the left needs some explanation. Roughly speaking, this

is because the values of Ck (q, a) for k=1 and k>1 are quite different. In fact,

when k=1 and (a, q)=1, we have •bC1(q,a)•b=•bƒÊ(q)•b=1. On the other hand, for

k>1, the value of •bCk(q,a)•b may be much bigger. We have (see [10], Chap. VI,

Problem 14b (a)), for any character x modulo q,

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where d(q) is the divisor function, and m a constant depending on k. On the other

hand,

It therefore follows that •bCk(q, a)•bwith k>1 usually takes values as big as q1/2, and this factor must be taken into consideration on the left-hand side of Theorem 1.1. Theorem 1.1 is non-trivial even in the extreme case Q=1, where ƒÆ can be as big as x1-k L-B. This can be compared with the range of •bƒÉ•b in (1.2).

The proof of Theorem 1.1 makes use of a large sieve estimate for a general

Dirichlet polynomial, which has previously been established in [4]; see Lemma 2.1

below. Hence our proof is different from those for the linear case in [8] and [5].

Just as the corresponding result (1.5) in the case k=1,

Theorem 1.1 is also a

consequence of GRH, and can be used as well in the additive theory of prime

numbers. For example, by Theorem 1.1 one can show that, if N is a large integer

and satisfies some necessary congruence conditions, then the Diophantine equation

with prime variables p1,... ,p5 is sovable for almost all q_??_N1/4-ƒÃ and all l with (l, q)=1. When q=l=1, this gives the classical result of Hua [3] for the quadratic

Waring-Goldbach problem.

2. Preliminaries and transformation of the problem

To prove Theorem 1.1, we will apply Heath-Brown's identity and then the

following Lemma 2.1. With the application of Heath-Brown's identity in mind, let

X2/5<Y_??_X and M1,...,M10 be positive numbers such that

(2.1)

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382 JIANYA LIU and JINGMEI YE

Then

we define the functions

and

where x is a Dirichlet character and s a complex variable. The following mean-value estimate for •bF•b is Lemma 5.2 in [4].

LEMMA 2.1. Let F(s, x) be defined as above. Then for any R_??_1 and T>0,

(2.2)

Here and throughout, ‡”*x mod r means the summation over all primitive characters x modulo r, and c denotes a positive constant whose value may change in different places.

3. Proof of the theorem

Introducing the Dirichlet characters, we have

(3.1)

where

and

with ƒÂx=1 or 0 according as x is principal or not. Note that Ck(x, a) reduces to Ck (q, a) if x mod q is principal. Therefore,

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(3.2)

where

x0 mod q is the principal

character.

For x mod r and x0 mod q in the last

line, we have

Therefore we can replace W(XX0, ƒÉ) by W(X, ƒÉ)+O(L2) in the last term of (3.2). By (1.7), the last term in (3.2) is bounded by

Therefore

Theorem

1.1 is a consequence

of the estimate

(3.3)

where R_??_Q.

The estimation of J falls naturally into two cases according as R is small or

l arge. For R>LC,

where C is some positive constant, one appeals to contour

integration, mean-value estimates for Dirichlet L-functions or their derivatives, the

large sieve inequality, and Heath-Brown's identity. While for R_??_Lc,

one uses the

classical zero-density estimates and zero-free region for Dirichlet L-functions.

We first establish the following result for large R. In Lemma 3.2 we shall

consider small R.

LEMMA

3.1. Let J be as in (3.3). Then for arbitrary A>0, there exists a

constant C=C(A)>0,

such that for

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384 JIANYA LIU and JINGMEI YE

we have

PROOF.

To the sum

(3.4)

we apply Heath-Brown's identity (see Lemma 1 in [2]) which states that

In (2.1) we take Y=x2/5 , and X=x; define aj(m), fj(s, x), and F(s, x) as before.

Therefore (3.4) is a linear combination of 0(L10) terms, each of which is of the

form

where M denotes the vector (M1, M2,.. . ,M10) with Mj as in (2.1). Note that

some of the intervals (Mj, 2Mj] may contain only the integer 1. By using Perron's

summation formula (see for example, Lemma 3.12 in [9]) and then shifting the

contour to the left, the above o (u; M) is

where

T is a parameter

satisfying

2_??_T_??x.

Take T=x.

The integral

on the two

horizontal

segments

above can be easily estimated

as

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Thus,

Since R>Lc(so x•‚x0), we have

and consequently W(X, ƒÉ) is a linear combination of 0(L10) terms, each of which is of the form

By changing

variables

in the inner

integral,

we deduce

from

the above

formulae

that

(3.5)

where the maximum is taken over all M=(M1,M2,...,Mlo). The contribution of ƒÆxkL12 to the J in (3.3) is

by the definition of ƒÆ in (1.6). This is acceptable if B is sufficiently large. Since

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(3.6)

where T0=4kƒÎxkƒÆ. Here the choice of T0 is to ensure that ~t + 2kirAvf > ~t~/2 whenever •bt•b>To; in fact,

It therefore follows from (3.5) and (3.6) that (3.3) is a consequence of the following

two estimates: For 0<T1_??_To, we have

(3.7)

while for T0<T2_??_T,

we have

(3.8)

By Lemma 2.1, the left-hand side of (3.7) is now

Since T0_??_ƒÆxk, the above quantity is acceptable provided that 0 satisfies (1.6) and R>Lc with sufficiently large B and C. This establishes (3.7).

By Lemma 2.1 again, the left-hand side of (3.8) is

which is acceptable provided that Lc<R_??_x1/2-ƒÃ with a sufficiently large C.

This establishes (3.8), and Lemma 3.1 now follows.

Now we treat

the case R_??_Lc.

LEMMA

3.2. Let A>0

be arbitrary and C=C(A)

be determined as in

Lemma 3.1. Let R_??_

Lc. Then there exists B=B(A)>0

such that

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We remark that the constant B in Lemma 3.2 is related to the definition of ƒÆ in (1.6).

PROOF. We use the explicit formula (see [1], Chap. 19, (6)-(7))

where ƒÏ=ƒÀ+iƒÁ runs over non-trivial zeros of the function L(s, x), b(x) a constant depending on x, and T_??_2 is a parameter. Take T=x. By partial summation,

(3.9)

The last integral in (3.9) can be estimated using an argument similar to the one

used for deriving (3.6) from (3.5). We get

where T0=4kƒÎxkƒÆ, the same as before. Inserting this into (3.9) and then taking summation over x mod r and r •` R_??_Lc, we have

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say.

We see that J0 is obviously acceptable. By (1.6),

which is acceptable if B is sufficiently large.

The term J2 will be bounded by Vinogradov's zero-free region (see

Satz VIII. 6.2 in Prachar [7]), which states that for any x mod r, there exists a constant c3>0 such that L(ƒÐr+it,x)•‚0 in the region

except for the possible Siegel zero. However, since r_??_Lc, the Siegel zero does not exist in the present situation. It follows that L(s, x) is zero-free for ƒÐ_??_1-ƒÅ(T)

and •bt•b_??_T, where ƒÅ(T)= c3/(21og4/5(ƒÁ+2)) for ƒÁ_??_0. Consequently, the inner

sum in J2 is

Therefore,

which

is also acceptable.

To bound J1, we use Ingham's zero-density theorem (see e.g. [6], Theorem 4.3

for a proof) that

where N(ƒÐ,ƒÁ, x) denotes the number of zeros ƒÏ=ƒÀ+iƒÁ of L(s, x) with ƒÐ_??_ƒÀ_??_1, •bƒÁ•b_??_ T. Therefore,

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Denote by f(T1, ƒÐ) the exponential function above, and we will analyze f (T1, ƒÐ) in detail.

Suppose first 4/5_??_ƒÐ_??_1-ƒÅ(T0), so that

From this and the zero-free

region,

it follows that

Secondly we consider 3/5_??_ƒÐ_??_4/5, which implies that

Since rT1_??_rT0<<rƒÆxk<<xL_B, we have log(rT1)_??_L+0(1), and consequently

Finally we deal with the case 1/2_??_ƒÐ_??_3/5. Now we have

and consequently,

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