On the Piatetski-Shapiro-Vinogradov theorem

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

A

NGEL

K

UMCHEV

On the Piatetski-Shapiro-Vinogradov theorem

Journal de Théorie des Nombres de Bordeaux, tome

9, n

o

_{1 (1997),}

p. 11-23

<http://www.numdam.org/item?id=JTNB_1997__9_1_11_0>

© Université Bordeaux 1, 1997, tous droits réservés.

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

On the

_{Piatetski-Shapiro-Vinogradov}

Theorem

par ANGEL KUMCHEV

RÉSUMÉ. Dans cet _article, nous considérons la formule _{asymptotique pour}

le nombre de _{représentations} d’un entier _impair N sous la forme p1 + p2 +

p3 = N, où les pi sont des nombres premiers du type pi =

[n1/03B3i] ;

nous

utilisons la méthode de van der _Corput en dimension deux et nous étendons

le domaine de validité de la formule _asymptotique en affaiblissant les

hy-pothèses sur les _03B3i. Dans le cas le _plus intéressant _03B31 = _03B32 = _03B33 = _03B3,

notre résultat entraîne que tout entier impair assez grand s’écrit comme

la somme de trois nombres _premiers de _{Piatetski-Shapiro} du type 03B3 pour

50/53

03B3 1.

ABSTRACT. In this paper we consider the _asymptotic formula for the

num-ber of the solutions of the _equation p1 + p2 + p3 = _N where N is an odd

integer and the unknowns _pi are _prime numbers of the form pi =

[n1/03B3i].

We use the two-dimensional van der Corput’s method _{to prove} it

un-der less restrictive conditions than before. In the most _interesting case

03B31 = _03B32 = _{03B33 = 03B3} our theorem implies that _every sufficiently large odd

integer N may be written as the sum of three _{Piatetski-Shapiro primes} of

type 03B3 for _50/53 03B3 1.

1. Introduction.

In 1937 I. M.

_Vinogradov

_[15]

solved the Goldbach

_ternary

_problem.

He

proved

that for a

sufficiently

large

odd

_{integer N,}

where

_C~(N)

is the

_singular

series

Key words and _{phrases. Piatetski-Shapiro primes,} Goldbach _{problem, exponential}

sums.

Manuscrit reçu le 26 novembre 1996

In 1986 E.

_Wirsing

_[16]

considered the

_question

of the existence of thin sets of

_primes

S such that _every

_{sufficiently large}

odd

_integer

is the sum

of three elements of S

_(the

set of

_prime

numbers S is called to be thin if

E

1 =

o(1f(x)) ).

Wirsing

proved

that there exists such a set of

_primes

px,pES

S with the

_property

that

The set of the

_{Piatetski-Shapiro}

_primes

of

_{type 7}

1

is a well-known thin set of

_prime

numbers. The

_counting

function of

P,

was studied

_by

a number of authors

[1,

_4-6,

8-14].

The best results are

given by

[14]

and

_[10]

where it is

_proved

that

for

_5302/6121

_q

_1,

and

for

_38/45

₁ 1.

In 1992 A.

_Balog

and J. P. Friedlander

_[2]

considered the

_ternary

Gold-bach

_problem

with variables restricted to

_{Piatetski-Shapiro}

_primes.

_They

proved

that if _{11, y2, 13} are fixed real numbers such

_{that -yi}

1

_{and Ii}

is close to

_1,

and N is a

sufficiently large

odd

_integer,

then the

_asymptotic

formula

₍₃₎

below is valid.

There are two

_{interesting special}

cases of this theorem.

1 in

(3)

we obtain an

_asymptotic

formula for the number of

_{representations}

of a

large

odd

integer

as the sum of three

_{Piatetski-Shapiro primes}

of

_type

y. If 11 = ₇₂ = 1 in

(3),

_we obtain an

_asymptotic

formula for the number

of

_{representations}

of a

_large

odd

_integer

as the sum of two

_primes

and a

Piatetski-Shapiro

prime.

The theorem of

_Balog

and Friedlander

_implies

that in the first case

the

_asymptotic

formula is valid for

_20/21

_{y 1,}

and in the second-for

_8/9

₁

1. J. Rivat

_[14]

extended _{the range}

_20/21

_y

1 to

188/199

1 ~ 1,

and C.-H. Jia

_[7]

used a sieve method to show that there exists a

_positive

constant po =

Po ( 1)

such that

for

_15/16

_-y 1.

In this paper we use better estimates of an

_exponential

sum to _prove: THEOREM 1. Let y2 1’3 be fixed real numbers such that 0 1 and

Denote

_by

_T(N)

the number of the

_{representations}

of the

_integer

N as the sum of three

primes

_{pl, p2, p3} such that _pi E

Py,.Then

the

asymptotic

formula

holds. Here

_6(N)

is defined

_by

_(2).

In the

_special

cases mentioned above this theorem

gives:

COROLLARY 1. For _any fixed

_50/53

_{-y -.5}

1 _every

_sufficiently

_large

odd

integer

may be written as the sum of three

Piatetski-Shapiro

_prime

numbers

of

_type

7.

COROLLARY 2. _{For any fixed}

_64/73

₇

1 _every

_{sufficiently large}

odd

integer

may be written as the sum of two

_primes

and a

Piatetski-Shapiro

prime

number of

_type

-y.

In both cases we _may obtain an

_asymptotic

formula for the number

of solutions. Thus Theorem 1

_improves

the known results of this

_type

contained in

_[2]

and

_[14].

Note also that

_64/73

= 0.8767 ... is _not much

2. Notation.

In this paper p, pl, ... are

_{primes; 6}

is an

arbitrary

small

positive

num-ber,

not _necessary the same in the different _appearances. The constants cl, c2, ... in Section 4

depend

at most on _7. We use

[x], fxl

and

llxll

to denote the

_integral

_part

of x, the fractional

part

of x and the distance from

x to the nearest

_integer

_{correspondingly. A(n)}

is von

Mangoldt’s

function;

e(x)

= = x -

M -

2.

f (x)

G

_g(x)

means that

f (x) = 0(g(x));

_I

g(x)

means that

f (~) « g(x)

G

_{f (x);}

f (xl, ... ,

Xn)

means that

for all

_n-tuples

_{( j 1, ... , jn )}

for which it makes sence.

x N X means that x runs

_through

a subinterval of

(X,

2X],

which

end-points

are not necessary the same in the different

_equations

and _may

depend

on the outer summation

_variables;

thus we _may

_write,

for

example,

We also define

3. Preliminaries.

The idea of the

_proof

of Theorem 1 is to reduce it to the

_following

The reduction

depends

on the

_asymptotic

formula for an

_exponential

sum

(Theorem

2

_below)

and indeed was done

_by

_Balog

and Friedlander in

Sec-tion 2 of

_[2].

_They

showed that

provided

that for 1 i 3 the estimate

holds with

₆₁

_{= ~(1 -}

₇₂₎

_{+ ~(1 -}

_y3),

₆₂

_{= ~(1 -}

_73).

and

₆₃

=

_0,

correspondingly.

Thus Theorem 1 follows from

_{(4), (5)}

and the

_following

improved

version of Theorem 4 of

_[2].

THEOREM 2. Assume that

_071,0~1-7

and

Then,

uniforrnl y

in a E

_{( 0,1 ) ,}

yve have

where the

_implied

constant

_depends

at most

_{on y, 6}

and c.

Throughout

the rest of this section we reduce the

proof

of Theorem 2

to the estimation of a double

_exponential

sum.

_Denoting

the sum in the

left-hand side of

₍₆₎

_by

_F(a),

and that in the

_right-hand

side

_by

_G(a)

we

get

It is easy to derive

₍₆₎

from this

_equality

and the estimate

using

the

_simplest

_splitting

_up

_argument,

we obtain that to _prove

₍₇₎

it is

sufficient to obtain

From here on x is a fixed

sufficiently

_large

number

_subjected

to x N. We recall the well-known

_expansions

where

We

_put

_Ho

= and

apply

the first

expansion

for the left-hand

side of

_{(8). Similarly}

to

_[4,

_p.246]

and

_{[2, p.51]}

we treat the error terms via the second and the estimate of van der

Corput

[3,

Theorem

2.2].

The

obtained estimate is admissible if

_{2(1 - y)}

+ 36

1,

so it remains to _prove that for each H

Ho

Working similarly

to

_{[4, p.247]}

we find that for to establish the last

_inequality

it is sufficient to _prove

Otherwise we treat the sums

involving e(hn’)

and

e(h(n+1)7)

_separately.

Thus in all the cases, it suffices to _prove the

_following

Assume further that H and u is either

_0,

or 1. Then

We _prove this

_proposition

in Section

_5,

and therefore

_complete

the

_proofs

of both Theorem 1 and Theorem 2. The

_{proof depends}

on the estimation

of some

triple exponential

_sums, which we estimate in the next section. 4.

_Exponential

sums estimates.

In this section we consider sums of the form

where

If the coefficients

_a(m)

and

_{b(n) satisfy}

the conditions

we denote the sum

_{by SI,}

and if

_{they satisfy}

_{the conditions}

we denote it

by

SII.

For

_SII

we use the estimate obtained in

_[2,

_Proposition

_2]:

LEMMA 1. Let N

_satisfy

the conditions

Then

For

_SI

we

give

a new estimate contained in the

_following

lemma which we _prove

_using

van der

_Corput’s

method as it is described in

_[3].

LEMMA 2. be

_subjected

to

an d N

_satisfy

the condition

Then

Proof of Lemma 2. Since for

_{"I, 8,}

N such that

the lemma is

_proved

in

_[2,

_Proposition

_3],

it is sufficient to consider the

case

In the rest of the

_proof

we _{suppose that the}

_inequality

₍₁₄₎

_{holds. We}

also note that the condition mn - z

implies

x.

We

_begin

with the remark that since

amn +

_h(mn

+

_u)"

= arnn + + +

one has

provided

that

_{1-, + 38}

1. Now we

_apply

the

_{Cauchy-Schwarz inequality}

and

_Weyl-van

der

_Corput

lemma

_[3,

Lemma

_2.5]

to the sum over n and

_get

where

_{f 2 (m, n)}

=

fi

(m, n + q) - f 1 (rn, n)

and

Q

N is _a

_parameter

_{at our}

which makes the first term in

₍₁₅₎

admissible.

_Applying

partial

summation to the sums over n and m

successively

we find that if N &#x3E;

X2(1--Y)+36+£

(which

is not restrictive in view of

_(13)),

then

We write k =

[aq]

and 0 =

{aq}

(note

that k and 0 do _not

depend

_{on m}

and _n, and 0 B

₁₎

and we derive from

₍₁₇₎

that

where

_{f3 (m, n)}

= Brrt ₊

hm7 ((n

+

ql’ - n7).

Hence

Now we

apply

the Poisson summation formula

[3,

Lemma

3.6]

over m. We

remove the

_arising

smooth

weights

via

partial

summation and

_get

where

and

and m runs

_through

an interval

[M1,

which

_endpoints

are monomials

of n such that

_{M1, Mi x}

FM-1.

We substitute this estimate in

₍₁₈₎

and

Note that if

FM-1 «

xl,

i.e. the summation over m in the last sum

is too

_short,

then the trivial estimate of the sums over m and n in

(19)

is sufficient to obtain the desired result. Thus we consider further the case

FM ‘1 »

x£. We will estimate the sum over _{m, n in} two different ways. Our first tool is van der

Corput’s

estimate as it is stated in

[3,

Theorem

2.9].

We

_change

the order of summation and

_apply

it with _q = 1 to the sum over n

(this

is

equivalent

to the use of the

_exponent

_pair

(.1, ’)).

Summing

up the obtained estimate we find that

for

Thus,

we _{may suppose further that}

Note that when

₍₁₂₎

holds this bound is

_always

smaller than

_(14).

We can also estimate the sum in the left-hand side of

(19) differently.

First we

apply

the

Weyl-van

der

Corput

inequality

over _n,

_introducing

in this way a new

_parameter

N. We

_get

which makes the contribution of the first term

_sufficiently

small.

Then we use the Poisson summation formula over n. Let

[Ni (m), N2 (m) ]

be the interval

_through

which runs r~ in

(21)

and let

[N3 (m), N4(m)]

be the

interval

_through

which runs when n runs

_through

~Nl (m),

_N2

(m)]

]

(then

N3 (m), N4 (m) x

qlF N-2).

Denoting by

Yn the

unique

solution of the

_equation

we obtain from

[3,

Lemma

3.6]

and

partial

summation that

where

If x£ then the first term in the

_right-hand

side of

₍₂₃₎

may be omitted at the cost ofax£ factor and the lemma follows from

_{(12), (13),}

(16),

(19)-(23).

We consider further the case xl. Then the

argument

of the

_proof

of Lemma 3.9 of

_[3]

shows that

Let us denote the sum over m, n in

(23)

by

T. We

apply

to it

_Weyl-van

der

_{Corput inequality}

over m and obtain

where

_Q2

FM-1

is a

_parameter,

(m, n)

runs

_through

a subdomain of

the domain of summation in

₍₂₃₎

and

We choose

and make the contribution of the first term in the

_right-hand

side of

₍₂₅₎

admissible. From

₍₂₄₎

and

₍₂₆₎

we derive that

where

A2

=

~l

₊

q2MF-1.

Now _we estimate the sum over _m,n via

_[3,

Lemma

_6.11~

with

_{(x, y) = (m, n)}

and we

_get

Combining

the last

_inequality

with

_(12),

_(13),

_{(16), (19)-(23), (25)}

and

(27),

we

_complete

the

proof

of the lemma.

5. Proof of the

_Proposition.

The inner sum in the left-hand side of

(13)

is an

_exponential

sum over

primes.

It is well-known that the sum

may be

decomposed

into double sums of two

_types-Type

I and

_Type

II

sums. Both

_Type

I and

_Type

II sums are sums of the form

mn-z

We call the sum

Type

I if the coefficients

a(m)

and

b(n)

satisfy

the

condi-tions

_(10),

and

_Type

II if

_{they satisfy}

the conditions

_(11).

We make the

_{decomposition using}

an

_identity

due Heath-Brown

_[4,

Lemma

_3]

LEMMA 3. Let 3 U V Z x and suppose that Z -

1/2

E

N,

x &#x3E;

_{64Z2 U,}

Z &#x3E;

_4U2,

V3 &#x3E;

32x. Assume further that

_F(n)

is a

_complex

valued function such that 1. Then the sum

may be

decomposed

into

x)

sums, each either of

Type

I with N &#x3E;

Z,

or of

_Type

II with U N V.

We

_apply

Lemma 3 with

_F(n)

=

e (an + h (n + u) -1),

U =

2-10xl-i+26+e,

V =

4~1~3

and

Then the

_Proposition

follows from Lemmas 1 and 2.

Acknowledgements.

The author would like to _{express his}

_gratitude

to Profs. D. I. Tolev and 0. T. Trifonov for their

_{encouragement.}

He would also like to thank the referee for his valuable

_suggestions.

REFERENCES

[1]

R. C. Baker, G. _Harman, J. _Rivat, Primes of the form

[nc],

J. Number Theory 50 (1995), 261-277.

[2]

A. _Balog, J. P. Friedlander, A hybrid of theorems _{of Vinogradov} and Piatetski-Shapiro, Pasific J. Math. 156 _(1992), 45-62.

[3]

S. W. Graham, G. A. Kolesnik, Van der _Corput’s Method _{of Exponential Sums,} L.M.S. Lecture Notes _{126, Cambridge University Press,} 1991.

[4]

D. R. Heath-Brown, The _{Piatetski-Shapiro prime} number _theorem, J. Number Theory 16

(1983),

242-266.

[5]

C.-H. Jia, On the _{Piatetski-Shapiro prime} number theorem _(II), Science in China Ser. A 36 _(1993), 913-926.

[6]

____, On the Piatetski-Shapiro prime number theorem, Chinese Ann. Math.

15B:1

_(1994),

9-22.

[7]

____, On the Piatetski-Shapiro-Vinogradov theorem, Acta Arith. 73

(1995),

1-28.

[8]

G. A. Kolesnik, The distribution _{of primes in sequences of} the _form

_[nc],

Mat. Zametki 2 _(1972), 117-128.

[9]

____, Primes of the form

[nc],

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[10]

A. Kumchev, On the distribution _{of prime} numbers of the form

[nc],

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[11]

D. Leitmann, Abschatzung trigonometrischer summen,, J. Reine Angew. Math.

317 _{(1980), 209-219.}

[12]

H.-Q. Liu, J. Rivat, On the _{Piatetski-Shapiro prime} number theorem, Bull. London Math. Soc. 24

_(1992),

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[13]

I. I. Piatetski-Shapiro, On the distribution _{of prime} numbers in sequences of the form

[f(n)],

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J. Rivat, Autour d’un theoreme de _{Piatetski-Shapiro, Thesis,} Université de Paris Sud, 1992.

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I. M. _{Vinogradov, Representation of} an odd number as the sum of three _primes,

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Angel KUMCHEV N. _{Vojvoda 20,} apt. 10 Plovdiv 4000

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