On the almost Goldbach problem of Linnik

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

J

IANYA

L

IU

M

ING

-C

HIT

L

IU

T

IANZE

W

ANG

On the almost Goldbach problem of Linnik

Journal de Théorie des Nombres de Bordeaux, tome

11, n

o

_{1 (1999),}

p. 133-147

<http://www.numdam.org/item?id=JTNB_1999__11_1_133_0>

© Université Bordeaux 1, 1999, 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/conditions). 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

On

the almost Goldbach

_problem

of Linnik

par JIANYA

LIU,

MING-CHIT

LIU,

et TIANZE WANG

RÉSUMÉ. On démontre _que sous GRH et _{pour k ~ 200,} tout

en-tier _pair assez

grand

est somme de deux nombres _{premiers impairs}

et de k _puissances de 2.

ABSTRACT. Under the Generalized Riemann

Hypothesis,

it is

proved

that for

_{any k ~}

200 there is Nk &#x3E; 0

_depending

on k

_only

such that _every even

_{integer ~ Nk}

is a sum of two odd _primes and k _powers of 2.

1. INTRODUCTION

In 1951 and

_1953,

Linnik

_[L1,L2]

_investigated

the

_following

"almost Gold-bach"

_{representation}

for even

_integers

N:

where

_{(and throughout)}

_p and _v, with or without

_subscripts,

stand for a

prime

and a

_{positive integer}

_{respectively.}

He showed that there is a constant

1~ &#x3E; 0 such that _every

_large

even

_integer

N can be written as

(1.1).

This result was

_{generalized by}

A.I.

_Vinogradov

_[Vi]

in several directions. In

_1975,

Gallagher

[G]

considerably simplified

the

_proofs

of Linnik and

_Vinogradov,

and established the

_following

result: For _any

_{integer k}

&#x3E; 2 there is a

_positive

constant

_Nk

_depending

on k

_only,

such that for each even

_integer

N &#x3E;

Nk,

where

_rk(N)

is the number of

_{representations}

of N in the form of

_{( 1.1 ) .}

Here

_log

N and

_{1092 N}

in

_(1.2)

_correspond

to the _{terms p} and 2v in

_{( 1.1 )}

respectively.

In the above results of

_Linnik,

_Vinogradov,

and

_Gallagher,

a

numerically

acceptable

value for 1~ still remains

_unspecified.

It is therefore not clear that how _{many powers} of 2 are needed to ensure

_rk(N)

&#x3E; 0. From

_(1.2)

we see that

_adding

more _powers of 2 does not

_change

the constant 2 in

The research is _{partially supported by Hong Kong} RGC research grant (HKU 7122/97P).

The first author is _{supported by} Post-Doctoral Fellowship of The _University of _{Hong Kong} and China NNSF Grant N° 19701019.

the main term N

_log2

N in

_(1.2)

but

_gives

a better error term

_only.

However,

due to the

_sparsity

of the sequence of powers of

2,

a desirable error term needs a

_large

number of them.

_Furthermore,

in view of the

Hardy-Littlewood theorem

_[HL]

on the

_exceptional

set of the Goldbach

_conjecture,

one _may

_anticipate

that even under the Generalized Riemann

_Hypothesis

(GRH)

a small k in

(1.1)

or

(1.2)

is not sufficient to

_give

the

_positiveness

of

Recently,

the _present authors showed that k = 770 is sufficient under

the GRH

_{[LLWI] ,}

and

_{unconditionally}

k = 54000 is

acceptable

[LLW2].

The _purpose of this _{paper is} to reduce the

_acceptable

value of k to 200

under the GRH. More

_precisely,

we shall _prove

Theorem 1. Assume the GRH. For _any

_integer

k &#x3E; 200 there exists a

positive

constant

_Nk

_depending

on k

only,

such that

if

N &#x3E;

Nk

is an even

integer

then

In

_particular,

each

_large

even

_{integer is}

a sum

of

two

_primes

and 200 _powers

of

2.

In a letter to

_Goldbach,

Euler

_asked,

and later answered

_by

himself

neg-atively,

the

_problem

of

_representing

each

_{sufficiently large}

odd

_integer

as a

sum of a

_prime

and a _power of 2.

However,

Romanoff

[R]

showed in 1934

that a

_{positive proportion}

of the odd

_integers

can be written in this _way.

And

_Gallagher

_[G]

_proved

that the

_density

of odd

_integers

_n, which can be

written as

tends to 1 as k - _oo, and from this he deduced his result

(1.2)

for the

almost Goldbach

_problem.

Unlike in

_[LLW1]

where we followed the above

approach

of

Gallagher

[G],

here we use Linnik’s

_original

idea

[LI,L2]

(see

_§4

below),

a lemma due to

Kaczorowski-Perelli-Pintz

_[KPP]

and

_{Languasco-Perelli}

_[LP]

_(see

Lemma 1

below),

and a well- known result of Chen

_[C]

obtained

_by

sieve methods

(see

our Lemma

3).

As

_usual,

_cp(n)

stands for the Euler

_function,

_ti(n)

the M6bius

_function,

and

_A(n)

the von

_Mangoldt

function.

Throughout

this _{paper, L}

always

stands for

₁₀₉₂

N.

_{Let X mod q}

_{and Xo}

mod _q denote a Dirichlet character

and the

_principal

character modulo _q

_{respectively.}

The letter C with

sub-scripts

denotes absolute _constants, and e denotes a

_positive

constant which

is

_arbitrarily

small.

The authors would like to thank the referee for useful comments and

suggestions.

2. THE MAJOR ARCS OF THE GOLDBACH PROBLEM

Let N be a

_{large integer,}

and

_P,

Q

_parameters

_satisfying

By

Dirichlet’s lemma on rational

_{approximation,}

each

be written in the form

for some

_{positive integers}

_{a, q} with

_{1 a q,}

_{(a, q)}

=

1,

and q

Q.

We

denote

_by

_{.M(a, q)}

the set of a

satisfying

(2.2),

and

_put

When

_q

P we call

A4 (a, q)

a

_major

arc. Note

that, by

(2.1),

all

_major

arcs are

_{mutually disjoint.}

Let

e(a)

=

exp(i27ra)

and

The _purpose of this section is to establish the

_following

Theorem 2. Assume the GRH. Let M be an even

_integer

with

NL-2

M

_N,

and

_specify

the P and

_Q

in

_{(2. 1)}

_{by putting}

Then we have

where

In the

_{Hardy-Littlewood method,}

the wider of the

_major

arcs will

_usually

give

the better results.

_Here,

we can see the influence of the GRH on the

width of the

_major

arcs. Under the GRH we can now widen the

length

of the

_major

arcs in

_(2.2)

_{by setting}

our

Q

=

N’12;

otherwise,

_we have

to

_considerably

narrow

them,

_{e.g. in}

[LLW2],

(4.1)

and

(2.1)

we set

_{Q =}

N179/180L-3.

Lemma 1. Assume the GRH. Let

_6x

be 1 or 0

according

as _X =

XO

_or not.

Then

uniformly for

any X mod q

and q

Q

M.

Proof.

This is

_[KPP,

Lemma

_1]

and the first

_paragraph

of

_[LP,§51.

Now we can

_give

Proof of

Theorem 2. Let

It suffices to _prove

since

_(2.5)

follows from

_(2.7)

via

_partial

summation.

Introducing

the Dirichlet characters

_(see

_[D,

_§26,

_(2)~),

we have

where

with

_{and 6x}

as in Lemma l.

_Thus,

Now we

_proceed

to estimate

_{h, I2}

and

I3.

Clearly,

Using

the

_{orthogonality}

of

_characters,

the bound

_{IT(X)12 ::;}

_q, and then Lemma

_1,

one

_gets

Applying Cauchy’s inequality

and then

_using

the

_elementary

estimate

one has

It therefore follows from

_(2.9)

that

Applying

the

_elementary

estimate

_(2.10)

_again,

one obtains

The contribution of the last 0-term to

_h

is

Since

we have

Inserting

(2.13)

and

_(2.14)

into

_(2.12),

we obtain

Therefore we conclude from

(2.8), (2.9), (2.11),

and

_(2.15)

that

By

(2.4)

and

NL-2

M

_N,

the last O-term is W

M log-1

M. This

3. AN EXPONENTIAL SUM OVER POWERS OF 2

The results of this section do not

_depend

on the GRH.

Let

Lemma 2.

_Letq

_1/(7e).

Then the set E

_of

a E

_{(0, 1]}

_for

which

(1 - q)L

has measure

L5/2 NB-1,

where

Proof.

This is

_[LLWI,

Lemma

_3].

Lemma 3. Let h be _any even

_integer,

and N

sufficiently

_large.

Then the

nurrtber

_of

solutions

_of

the

_equation

h =

pi - P2 with pj N is

uniformly for

all even

_integers

h =A

0. Here

Co

is as in

(2.6).

Proof.

This is

_[C,

Theorem

_3].

Lemma 4. We have

where

_C1

24.4189.

The above

_integral

is

_trivially

»

N,

so the _upper bound in

(3.1)

is of correct order of

_magnitude.

The

_inequality

_(3.1)

with an

unspecified

constant in the upper bound was obtained

_by

Romanoff

_[R]

_(see

also

[P],

Satz 8.1 on

p.173).

Therefore

Romanoff

_completed

the

_qualitative

estimate for the

_integral

in

_(3.1)

which

is,

in

_fact,

the sum of _squares of the

representations

of those

integers

of the

form _{p +}

_2’,

with _p and v in suitable _{ranges. From} this he

deduced,

by

Cauchy’s inequality,

that a

_{positive proportion}

of the odd

integers

can be

written as _{p +} 2’. What we do in our Lemma 4 is to obtain a

_quantitative

result of Romanoff’s

_inequality,

i.e. a numerical constant in the upper bound

in

_(3.1).

satisfying

(3.3)

Then we have

By

Lemma 3 one

has,

uniformly

for even

integer h,

where

_Co

is as in

_(2.6),

Thus for fixed _{ml, m2} with m1

~

m2, one sees that

If m2 &#x3E; m 1, then

Similarly,

for m2 ml. Since for fixed

one has

Also, by

the

_prime

number

theorem,

the number of solutions of

_(3.2)

with m1 =

if N is

_{sufficiently large.}

This in combination with

_(3.6)

_gives

The sum on the

right

hand side of

(3.7)

can be transformed as

where for odd

_d,

denotes the least

_{integer (} 2:}

1 for which 2e -

_1(mod

d).

Hence

_(3.7)

becomes

By

[HR,p.1281,

Co

&#x3E; 0.6601. Also

_by

_(3.5),

a

_{straightforward}

_computation

gives C2

7.8342 x 1.8998.

Now we

proceed

to estimate

_C3.

For

_{positive integers x,}

_put

Then for x =

1, 2, ..., 9,

_we have

To bound

_c(x)

for x &#x3E;

_10,

we let

x &#x3E;

_9,

, and prove that when

where 7

is Euler’s constant

_(so

1.7810 e7

_1.7811).

In

_{fact, according}

to

_[RS,(3.42)],

for _every d &#x3E; 3 we have

If x &#x3E;

_9,

then we have

and

_(3.9)

follows. Thus if x &#x3E;

_10,

then

Obviously,

this estimate also holds for

_non-integral

x &#x3E; 10.

We therefore conclude that

Consequently,

(3.8)

becomes

with

as stated in the lemma. This in combination with

(3.4)

gives

(3.1).

The

4. PROOF OF THEOREM 1

Lemma 5. Assume the GRH. Let

_{P, Q}

be

_{defined as}

in

_{(2. 1).}

Then

_for

a ~

Proof.

This is a _consequence of

[LLW1,

Lemma

_1].

Lemma 6. Let

_tk(n)

be the number

_of

solutions

_of

the

_equation

n = 2"1 +

Proof.

The first sum under consideration is the number of k-dimensional vectors

_{(vl, ..., Vk)}

such that 2v1 + 2V2 + ... + 2vk N.

Thus,

and the first estimate follows. The second estimate can be established as

follows:

This

_completes

the

_proof

of Lemma 6. Now we

_give

Proof of

Theorem 1. Let

_{tk (N - M)}

be as in Lemma

6, S

as in Lemma 2

Now we estimate

Ji , J2, J3

in

(4.1)

_{respectively.}

-2

To estimate

_J1,

we bound the contribution from M

lV L-2

as follows:

on

_using

the first estimate of Lemma 6. The contribution from other M can

be estimated

_by

Theorem 2 and the second estimate in Lemma

_6,

which

give

provided

k 2:

2 and N &#x3E;

_Nk,é.

Thus &#x3E; 2 and N &#x3E; _then

To estimate

_J2,

one notes that

By

Lemma

_5,

we have

S(cY, N)

«

N3/41og2

N for a E

_C(M).

Let ₁₇ =

0.0161 so that the definition of e in Lemma 2

gives

O 0.4998

1~2.

On

_using

Lemmas 2 and

_4,

the last

_integral

_J3

can be estimated as

Inserting

(4.2), (4.3)

and

_(4.4)

into

_(4.1),

we

_get

if k &#x3E;

2 and Also when 200 and E =

10-6,

_one has

C1 (1

-r)k-2

+ 3E 0.9818.

_Consequently

if k &#x3E;

200 and

_Nk,

then

_(4.5)

becomes

on

_recalling

that

Co

&#x3E; _{0.6601. This proves}

_(1.3)

and hence Theorem 1.

REFERENCES

[C] J.R. Chen, On Goldbach’s problem and the sieve methods. Sci. Sin., 21 _(1978), 701-739.

[D] H. _{Davenport, Multiplicative} Number Theory. 2nd ed., Springer, 1980.

[G] P.X. _Gallagher, Primes and powers of 2. Invent. Math. _29(1975), 125-142.

[HL] G. H. _Hardy and J. E. Littlewood, Some problems of "patitio numerorum" V: A further

contribution to the _{study of} Goldbach’s _problem. Proc. London Math. Soc. ₍₂₎ 22 _(1923), 45-56.

[HR] H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, 1974.

[KPP] J. Kaczorowski, A. Perelli and J. _Pintz, A note on the exceptional set _for Goldbach’s

problem in short intervals. Mh. Math. 116 _(1993), 275-282; corrigendum 119 _(1995), 215-216.

[L1] Yu. V. Linnik, Prime numbers and powers of two. _Trudy Mat. Inst. Steklov 38 _(1951), 151-169.

[L2] Yu.V. Linnik, Addition of prime numbers and powers of one and the same number. Mat.

Sb. _{(N. S.)} 32 _(1953), 3-60.

[LLW1] J.Y. _Liu, M.C. _Liu, and T.Z. Wang, The number of powers of 2 in a _{representation of}

large even _{integers (I).} Sci. China Ser. A 41 (1998), 386-398.

[LLW2] J.Y. _Liu, M.C. _Liu, and T.Z. _Wang, The number of powers of 2 in a _{representation of}

large even _{integers (II).} Sci. China Ser. A. 41 (1998), 1255-1271.

[LP] A. _Languasco and A. _Perelli, A _pair correlation hypothesis and the exceptional set in Goldbach’s _problem. Mathematika 43 _(1996), 349-361.

[P] K. _{Prachar, Primzahlverteilung. Springer,} 1957.

[R] N.P. _{Romanoff, Über einige} Sätze der additiven Zahlentheorie. Math. Ann. 109 _(1934), 668-678.

[RS] J.B. Rosser and L. _{Schoenfeld, Approximate formulas for} some _{functions of prime}

num-bers. Illinois J. Math. 6 _(1962), 64-94.

[Vi] A.I. _Vinogradov, On an "almost _{binary" problem.} Izv. Akad. Nauk. SSSR Ser. Mat. 20

Jianya Liu Department of Mathematics Shandong University Jinan,Shandong 250100 P.R. China E-mail : jyliu(Dsdu.edu.cn Ming-Chit Liu Department of Mathematics The _University of _{Hong Kong}

Pokfulanm,Hong Kong matmcliu0hkucc.hku.hk Tianze WANG Department of Mathematics Henan _University Kaifeng, Henan 475001 P.R. China