Trigonometric sums over primes III

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

G

LYN

H

ARMAN

Trigonometric sums over primes III

Journal de Théorie des Nombres de Bordeaux, tome 15, n

o

3 (2003),

p. 727-740.

<http://www.numdam.org/item?id=JTNB_2003__15_3_727_0>

© Université Bordeaux 1, 2003, tous droits réservés.

L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/legal.php). Toute uti-lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

1. Introduction

Throughout

this _paper we write

e(x)

=

exp(2xiz)

and let _p denote a

prime

variable. Sums of the form

arise in

_applications

of the

_{Hardy-Littlewood}

circle method to the

Waring-Goldbach

problem

(for

example

see

[9]),

or in

_problems

_involving

the dis-tribution of modulo one

(see

[1,

12]).

The aim of this _paper is to

_give

improved

bounds for these sums when

1~

&#x3E; 5

(although

Theorem 3 does

give

a better bound in certain _ranges than _any

_{previously published}

ex-plicit

result for k = 4 _as

well).

The first unconditional bounds for these

sums were

_{given by Vinogradov}

[10, 11].

The case k = 1 _was

required

for

his celebrated

_proof

of the

ternary

Goldbach

problem.

In 1981 the author

[4]

showed

_that,

if

then,

for k &#x3E; 2

1

with q

=

41-k, f

_&#x3E;

_0,

and _constants

implied

by

the « notation will

de-pend

at most on k and e. We shall state this

_{dependence explicitly}

in the statements of theorems and lemmas. This

_improved

the

_{value 1}

= +

1))-l

obtained

_by

_Vinogradov,

and

_generalised

the case k = 2

given by

Ghosh

_[3].

_Improved

bounds for

were

_given

in

[5].

_Then,

in

_[1],

it was shown

(essentially)

that

_{if a = :}

with

k

q

q near P 2 we have

with

_{a(k) =}

_{(3.2k-1 )-1.}

This

_bound,

in which the

_exponent

is

_only

a factor of 3 _away from the classical bound for a

Weyl

_sum, was used to _prove new results on the distribution of

a~

modulo one. The new idea introduced for this result was to estimate double sums

by approximating

Qmk

_{(or i (mt -}

?7))

with another

_{rational §}

where one

Q Q 2

could control the behaviour of s

_{= s (m) (or s (ml, m2 ) )}

on _average over m

(or ml, m2).

Since then various authors have

_given

more

_general

estimates for these sums

_using

_{similar ideas. For technical} reasons it is often better in these results to _{restrict the range of summation} to

_{p - P}

where a N A

throughout

this paper means A a 2A. The culmination of this work is Lemma 3.3 of

_[7],

which we state as follows.

Theorem 1.1. Let

_4,,e

&#x3E; 0 and let P E

_I~,

P &#x3E; 2.

_Suppose

that a is a real

number,

and there exist

_integers

_{a, q} with

Then one has I

In the above is the

_{multiplicative}

function whose value for a

_prime

power

pc

is

given by

This

_gives

_{wk (q)}

_{« q-l/k,}

_while,

on _average,

q-1/2

is nearer the truth.

Theorem 1.1 was

_applied

to obtain new results for the

Waring-Goldbach

problem.

The _purpose of this note is to

_improve

the first term on the

_right

hand side of

_(5).

We are also able to consider more relaxed conditions on a. _Our main result is as follows.

Theorem 1.2. Let k E

_{~, k &#x3E;}

_{5, E}

&#x3E; 0 and let

_{P, H}

_2,

Suppose

that a is a real

number,

and there exist

_integers

_{a, q} with Then

The restriction to k &#x3E; 5 is caused

_by

_parts

of the

_proof

which

_require

a(k)

(8k

+

_2)-l.

For

_larger

k one should be able to obtain an

_exponent

a(k) =

(4.5J~2 log 1~)-1

by using

[13]

in

_place

of

_{Weyl’s inequality}

_(used

im-plicitly

in Lemma 3.1

_below).

As

applications

of this result we note that the case k = 5 leads to a

_{major simplification}

of the

_proof

of Theorem 2 in

_[7],

_and,

for

_general

k &#x3E; 5 this result should lead to

_improved

bounds for

_{solving inequalities}

_involving

_powers of

_primes.

The main factor in our

strengthening

of Theorem 1 is our estimation of

_Type

I sums

(Lemma

2

be-low),

which in some _{ways is less}

sophisticated

than the

corresponding

result

in

_[7],

but the

_injection

of a

_simple

idea from

[1]

_enlarges

the

_permissible

range of

parameters.

Before

_stating

our final result we will describe what is known when a is rational. It turns out that one such result combines well with Theorem 1.2

to

_produce

a

_strong

bound for

_{trigonometric}

sums over

_primes.

In

[3]

the author remarked that the bound

follows from work of

_Vaughan

_[8].

This is non-trivial for _q and

_greatly

_improves

on Theorem 1.2 for small _q.

Fouvry

and Michel

[2]

proved

quite

a

_general

result which

_shows,

as one

_{example, that, for q}

a

prime,

q &#x3E;

_{P, ~ &#x3E;}

_2,

I I

This result is

_only

non-trivial for the short _range P _q

P6 ,

but it

improves

our results in a narrow _range. It is

interesting

to note that the

exponents

of both the above results for rational a are

_independent

of k. The bound

₍₉₎

can be combined with Theorem 1.2 to

_produce

the

_following.

Theorem 1.3.

_Suppose

that k &#x3E;

_5,

P &#x3E;

_2,

E &#x3E; 0 and a E R such that

₍₂₎

holds. Then

Here

Moreover,

the bound

₍₁₀₎

remains valid

_for

k = 4 with

a(k)

replaced by

i2.

Remark. This

_improves

on all

previous

bounds,

not

_{just by}

virtue of

im-proving

the term

_independent

of _q, but also in the way the bound behaves

for small or

large

_q.

Indeed,

for very small or _very

large q,

(10)

_gives

a better bound than can be obtained for the standard

_Weyl

sum. The fact that

_{Tl (k)}

and

_{T2 (k)}

do not tend to zero with

_increasing

1~ is _very unusual.

Using

Theorem 1.2 alone we would

_only

obtain

Tl (k)

=

T2 (k)

=

2~ .

The

explanation

for this

_phenomenon

would seem to lie in the

_underlying

model

complete

sum with which the

_exponential

sum is

_compared,

either

for which we know _very different bounds hold in

_general.

Proof.

(Theorem 1.3)

If

Pi

_q

P 2

the result follows

_immediately

from Theorem 1.2. Otherwise we can

_pick

_{u, v,} with

If v &#x3E;

P 4

the result

_again

follows from Theorem 1.2 since 8k

_{_ ~ (k) ~ 1.}

We now _suppose that v

Pi .

_{Writin g}

we can _suppose that S »

Pl-o’+E.

Let R = 1 +

pkla -

_{Then, by}

(9)

while

_{(8) (or (5)}

if k =

4)

gives

From

₍₁₁₎

and

₍₁₂₎

we deduce

that,

for _any

~i

E

_~0,1~,

1

We first consider the case q P 2 . Now we cannot have v

q/2

for

otherwise we

_get

the

_{contradictory}

_sequence of

inequalities:

We choose

_{3

_{= ~}

in

_{(13) (to}

eliminate

_R)

and then

This establishes

₍₁₀₎

for this case, so we henceforth suppose

that q

&#x3E;

P§ .

From the first case

(with

_q

_{replaced by}

_v)

we _may also _suppose that v

and so v It follows that

(noting q

&#x3E;

p2

&#x3E;

2Pi

&#x3E;

2v)

Hence vR »

_pkq-1.

We now choose a =

3k/(4k -1)

in

(13)

(to

equalise

the _powers of R and

v)

and deduce that

This establishes

₍₁₀₎

for this case and

_so.completes

the

_proof.

0

2. Theorem 1.2 deduced from 3 lemmas

We write

_A(n)

for the von

Mangoldt

function.

By

a familiar

_argument

(partial

summation)

it suffices to show

_that,

for

P’ -

_P,

An

_application

of Heath-Brown’s

_generalised

_{Vaughan identity}

_[6]

as on

page 172 of

[1]

gives

the

_following

_{decomposition}

of the left hand side of

(14).

In the

_following

we write

Tk ( n)

for the number of _ways of

writing n

as a

product

of k

_{positive integers}

with the convention that

rl(n) =-

1.

_Also,

Lemma 2.1. Let 0 _’Y

₁₂

and _suppose that we have the

Type

I estimate

12

, and the

Type

II estimate

_ _

The

_{following result, proved}

in section 4 contains the main

_improvement

on

[7].

Lemma 2.2. Let e &#x3E; 0.

_{Suppose H, P,}

M are

positive

reals with 2

P,

(6)

holds,

and

Let a be a real number such that there exist

_integers

_{a, q} such that

(7)

holds.

Then,

« we have

Remark. In Lemma 3.2 of

_[7]

a trilinear sum is considered and a bound is

given

which is

_stronger

than

_(20).

_However,

we can take M

_larger

than in

[7]

and this enables us to utilize the result of Lemma 2.1. Our

_advantage

is thus obtained

_{by establishing}

a weaker result for a

_larger

_{range of}

_parameters.

The reader will see that it is

_possible

to

_improve

the first term on the

_right

hand side of

_(20),

but this would

_complicate

the

_proof

_yet

not

_strengthen

the result of our main theorem.

Our final

_component

in the

_proof

of Theorem 1.2 is the

_{following slightly}

revised version of Lemma 3.1 in

_[7].

The

_proof

will be

_given

in section 3.

Lemma 2.3. Let c &#x3E; 0.

_Suppose

_H,

_P,

M are

positive

reals with

Suppose

that am,

bn

satisf y

(1 ~’)

and a, a, q

satisfy

(7).

Then,

for

P’ ,

P

Proof.

(Theorem 1.2)

We

_apply

Lemma 2.1

_{with y}

=

a(k).

We _{can use}

Lemma 2.2 to estimate

₍₁₅₎

and Lemma 2.3 is

_applicable

to

_{(16) (replacing}

c

_by

e/2

in those

lemmas).

This establishes

(18)

with

and so

(8)

follows. 0

3. Proof of Lemma 2.3

First we need an

important

lemma which is a more

_general

version of Lemma 2.1 in

_{[7] (where}

_p is fixed as

21-k)

but which follows in the same

way from Lemmas

2.4,

6.1 and 6.2 of

[9].

Lemma 3.1. Let _p E

_(0,

21-k~, e

&#x3E;

_{0, X &#x3E;}

1.

_Then,

either

or there exist

integers

_{a, q} with

and

Proof.

(Lemma 2.3)

We need to make two

_changes

to the

_proof

of Lemma 3.1 in

_[7]:

first to allow for the

_changed

conditions on _{a, a, q;} second to

allow for the

_changed

conditions on _am,

_bn.

We outline the

_steps

in the

proof

to

_help

the reader follow the

_argument.

Write

We

_begin

with a standard

_{application of Cauchy’s inequality}

to the left hand side of

₍₂₀₎

_together

with a familiar

shufliing

of the orders of summation:

Here N =

P/2M

and

The terms with _{nl - n2} can

_only

be dealt with

trivially

and this leads to

a

(PM)2+f

term which is « in view of the upper bound for M

contained in

_(21).

We then assume that the sum over m above is _&#x3E;

M pf /T

and this

_{shows, by}

Lemma

_3.1,

that we can

_approximate

(n -

_{2 1}

_by

a

fraction

where

In the

_following

we

closely

follow

[6]

and assume for the moment that in

place

of we have

bn

satisfying

the condition

(bn

«

_logn)

which

occurs in that _paper. We take out the

factor

_no =

(ni, n2),

_{put n}

=

nl/np,

1 =

(n2 - nl)/no

and write

Terms with _no &#x3E; TP-1 can

clearly

be

neglected,

so we henceforth _suppose

no

TP-’.

_Following

the

_argument

on _{page 11} of

[7]

we obtain a suitable

bound,

except

possibly

when

_(see

_(3.13)

in

_[7])

We deduce from Dirichlet’s

_{approximation}

theorem that there exist d E Z

As in

_(7~,

the sums to be estimated are of a suitable

_size,

unless

(see (3.18))

Now the conditions on _{a, a, q} are

only

needed in

[7]

(see

_page

13)

to show

that

_(see

the

_display

after

_(3.19)),

In our situation we have

from

_(21).

Hence

₍₂₃₎

is satisfied in our case.

Now we deal with the

_changed

conditions

on bn

which,

with the altered

hypothesis

for

_eventually

lead to the

_{higher log}

_powers in our result. This means that we must include a factor in

81

on _page 9 of

[7].

Now,

if

_{q &#x3E;}

P’ we can use «

qE/2

and the

_proof

_goes

_through

exactly

as in

[7].

If q

P" then we must

_persist

with this factor. Thus the

inequality

(3.8)

in

_[7]

_becomes,

upon

writing

No

=

N/no,

by

making

a suitable modification to the

_argument

on _page 10 of

[7]

to

deal with the

_Tv(n)

factor. The factor is then carried

_through

the

argument

until at last one has to estimate the sum

(see

the un-numbered

1

However,

since

_T1J(no)

«

_n’

this sum is

by modifying

Lemma 2.3 of

_[7].

_Combining

the results we obtain

, n

Here h =

2 + v2 + u2

and so the worst case has u =

1, v

= 5

(or

vice

versa)

giving h

= 28 _as

required

to establish

(22).

0

4. Proof of Lemma 2.2

First we need a further two lemmas.

Proof.

We _may assume that f

1.

We

have,

since

Zv~ (r) &#x3E; r- 2 ,

by

Lemma 2.4 of

_[7].

Lemma 4.2.

_{Suppose q,}

_E, 9 &#x3E;

_0,

d &#x3E;

_2,

M &#x3E; 1.

_Say

Then, given

r

d,

the number

of

solutions to

Proof.

This

_simple

bound follows

_by

the method of Lemma 6 in

_[1].

To see

We then need to count the number of solutions to

This number is

which

_gives

_(26).

0

Proof.

(Lemma 2.2)

We start in a similar manner to the

_proof

of Lemma 3.2 in

_[7].

Write

_Q

=

By

Lemma 3.1 with X =

P/m

and _p defined

by

X1-p -

pl-cr(k) /m,

for each _m, either

or there exist

_{integers b,}

r with

Here we have

_changed

a term P-" to

₂

since we _may assume that P is

_{sufficiently large,}

and we

require

an

_explicit

constant less than 1 later

in the

_{proof. Evidently}

we can

_ignore

those m for which

(27)

holds and

concentrate our attention on those m for which we obtain

(28).

For each

r

_Q

let

_,M(r)

be the set of all m - M for which

(28)

holds. Then

Now,

by

Lemma

_3.1,

Clearly

we can absorb the contribution from

_pairs

_{(m, r)}

with

into the first term on the

right

of

(29).

So henceforth we assume

(30)

is violated and denote the

_{corresponding}

sum

_by

S2.

Now there exist _c, d with

Moreover,

if d

_H/2,

we have

and so d =

q, c = _a in this _case. Now write

where

_{M(r, e)}

counts those m E

_{M (r)}

with

_{(m, d)}

= _e.

Clearly

we can

dispense

with those e

satisfying

e &#x3E; We now

_split

the

_argument

into two cases.

(i)

ekr

d. We can

apply

Lemma 4.2 to show that

The contribution to

_S2

from case

(i)

is therefore

where we have used Lemma

4.1,

M

P 2 +~~~~,

and d We note that we also

required

for the final

_inequality,

and it is this

_point

which limits our method to the case k &#x3E; 5.

(ii)

ekr &#x3E;

d. For this case we have d

!Q2

and thus d = _q, c = d. Also

Hence =

q,

and so

wk(r)

=

wk(q/(mk,q)).

We _can therefore estimate

this case as

We now consider

from Lemma 2.3 in

_[7].

On the other

_{hand, for q}

ME we first define the

multiplicative

function

_vk(n)

_by

for u &#x3E; 0 and 1 v k. We then have

by

Lemma 2.2 of

_[7]

since We thus have an estimate

in this case.

_Combining

_(i)

and

_(ii)

then establishes

₍₂₀₎

and

_completes

the

Acknowledgement.

The author thanks the referee for his comments. References

[1] R.C. _BAKER, G. _HARMAN, On the distribution _of

_03B1pk

modulo one. Mathematika 38 _(1991), 170-184.

[2] E. FOUVRY, P. _MICHEL, Sur certaines sommes _{d’exponentielles} sur les nombres premiers.

Ann. Sci. Ec. Norm. _Sup. IV Ser. 31 _(1998), 93-130.

[3] A. GHOSH, The distribution of

03B1p2

modulo one. Proc. London Math. Soc. ₍₃₎ 42 _(1981), 252-269.

[4] G. HARMAN, Trigonometric sums over _primes I. Mathematika 28 _(1981), 249-254.

[5] G. HARMAN, Trigonometric sums over _primes II. _Glasgow Math. J. 24 _(1983), 23-37.

[6] D.R. HEATH-BROWN, Prime numbers in short intervals and a _{generalized Vaughan identity.}

Can. J. Math. 34 _(1982), 1365-1377.

[7] K. KAWADA, T.D. WOOLEY, On the Waring-Goldbach problem for fourth and _{fifth powers.} Proc. London Math. Soc. ₍₃₎ 83 _(2001), 1-50.

[8] R.C. VAUGHAN, Mean value theorems in _prime number _theory. J. London Math. Soc. ₍₂₎ 10 _(1975), 153-162.

[9] R.C. _VAUGHAN, The _{Hardy-Littlewood} Method second edition. _{Cambridge University Press,} 1997.

[10] I.M. _VINOGRADOV, Some theorems _concerning the theory of primes. Rec. Math. Moscow N.S. 2 _(1937), 179-194.

[11] I.M. _VINOGRADOV, A new estimation _{of a trigonometric} sum _{containing primes.} Bull. Acad. Sci. URSS Ser. Math. 2 _(1938), 1-13.

[12] K.C. _WONG, On the distribution

_{of 03B1pk}

modulo one. _Glasgow Math. J. 39 (1997), 121-130.

[13] T.D. WOOLEY, New estimates _{for Weyl} sums. _Quart. J. Math. Oxford ₍₂₎ 46 _(1995), 119-127.

Glyn HARMAN

Royal Holloway University of London

EGHAM, Surrey. TW20 _OEX, UK E-mail :