On the greatest prime divisor of quadratic sequences

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

J. P

OMYKAŁA

On the greatest prime divisor of quadratic sequences

Journal de Théorie des Nombres de Bordeaux, tome

3, n

o

_{2 (1991),}

p. 361-375

<http://www.numdam.org/item?id=JTNB_1991__3_2_361_0>

© Université Bordeaux 1, 1991, tous droits réservés.

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

On the

_greatest

_prime

divisor

of

_quadratic

_sequences

par J. POMYKALA

1. Introduction. Let

_PT

be the

_greatest

_prime

divisor

where a is not a

_perfect

_square. In 1967 C.

_Hooley

(see[4])

showed that

Px

&#x3E;

_{where 1}

= i

and 6’ is _any

_positive

number. In 1982 J.-M. Deshouillers and H. Iwaniec

_applied

the new estimates of

_exponential

sums

of Kloosterman

_type

_(see[2])

to obtain the

_improvement

in the

_exponent

(see[I]) 1 =

10 = 1.2024...

(in

fact

they

considered the case a =

-1).

Assuming Selberg’s eigenvalue

conjecture

their method

_implies

the value q =

2 -

1.2247... One can avoid

_Selberg’s

_conjecture

_dealing

with

ad-ditional

_averaging

of Kloosterman sums over the levels of the Hecke

con-gruence

subgroups

involved here. This motivates the

investigation

of the

greatest

prime

divisor of

where IS is a rather thin set of

_primes

and _very

_general.

_Assuming

the

density

condition

we denote

by

the

_greatest

prime

divisor of

(1).

In such

circum-stances the _{progress in the}

_{exponent y}

is obtained for the values

Moreover the limit of the method is attained for e

_{= 2 -}

1 as

{3 -

1. In

the _{above range of variables}

_,Q,

O we define the function

We shall _prove the

_following

THEOREM.

₍₃

_1,

and a = be a

(unique)

solution of

FfJ,e

= 2 .

Then for _y =

0)

satisfying

the

equality

q = _sup and _any 6’ &#x3E; 0 we have

Remark.

The

_uniqueness

of

₀₎

is

_{implied by}

the

_integral

_{representation}

₍₂₀₎

of Moreover

_comparing

the values of

_D(x,

_P,

(3,

8)

with the values D = and D =

xt-E (see[l])

we obtain

2. Notation.

_Throughout

the _{paper - will be}

_{arbitrary sufficiently}

small

_positive

constant not

_necessarily

the same in each occurrence. More-over we take the

_following

notation:

e(z) -

the additive character

e2riz

f -

the Fourier transform

_{of f , i.e.,}

11/11-

means the _sup norm of

_f

~2013

means ~

(mod 1)

where ad -

_l(c).

S(a,

b;

c) -

means the Kloosterman sum i.e.

means the summation over residue classes a

(mod s)

with

1 X [ -

means the

cardinality

of the set X

J := t9

_{(mod m)-}

stands for a solution of the

congruence t9’

_{+ 1} -

0(m)

e(m)

:=

ft9 (mod m) :

~92

_~.1-

0(m)} I

p, q - denote

always

the

prime

numbers and q E B. 3.

_{Tchebyshev’s}

method.

Given O E

_{(0, 1)}

we consider the _sequence

the elements of which are counted which the smooth functions

g(n), B(q)

(drawn below)

with derivatives

where the constant

_implied

in the

_{symbol « depends}

on v

_only.

where

_A(d)

is von

_Mangoldt

_{function. Since 2 1}

log(2X2)/ logp

in the above and

where

we obtain that the _{error term}

_{0( . )}

_{contributes to}

T(x)

at most

Hence

By

the Poisson summation formula and the

_inequality

_q it is

_equal

to

To estimate the

_partial

sum

_T*(x)

=

M6bius

_function)

Letting

m = _p we obtain that for e = 1 the

corresponding

contribution to

T*(x)

is bounded

_by

In case e = 0

applying

the Poisson summation formula for the inner sum

in

₍₃₎

we

_get

-when

_{integrating f}

_(twice)

by

parts.

Hence the main con-p

tribution to

_T*(x)

is

_equal

to

Gathering

(2), (4), (5)

together

we obtain

Now the main

_goal

is to find the

_{largest possible}

value of

_Px

such that the _upper bound for

_S(x)

be less than

4.

_Application

of the sieve method.

Like Iwaniec and Deshouillers in

_[1]

we use the smooth

_partition

of

_unity

to

_split

the sum

S(x)

into J

_{2log x}

sums of the

_type

where

and such that

We estimate the

_typical

sum

using

an _upper bound sieve of level A

_{(1 ~}

A

x).

_By

₍₃₎

and

The value h = 0 contributes to the main

_term,

_namely

Denoting

the above sum is

_equal

to

provided

D x which we henceforth assume.

Therefore we obtain

The remainder term

_{(corresponding}

to the

_{values h # 0)}

will be considered in the next sections.

°

5. Transformation of the remainder term.

With the aid of the M6bius function

_(cf.(3))

we translate the condition

(q, m)

= 1 in the

_right-hand

side of

(7)

_to obtain the _sum

I

Applying

the smooth

_partition

of

_unity

we _may assume that the ranges of d

and h are controlled

by

the smooth functions with their

_supports

contained

in

_{~D, 2D~}

and

_{[H, 2H]}

_respectively

_(D,

H &#x3E;

_1).

Since the

_partial

integra-tion allows us to truncate the series

_dq’,

_P)

at

_lhl

we

shall assume that H The

oscillatory

character of the inner sum in is to be

_{expressed by}

means of the

following

LEMMA 1 congruence

is soluble then m is

represented properly

as a sum of two squares

There is a one to one

correspondence

between the

incongruent

solutions of

(10)

and the solutions

_{(r, s) of (11)}

_{given by}

PROOF. - see

[6]

and

[3],

p.34,

eq.

(68).

We use Lemma 1 to _express in terms of r and s.

Apply-ing

the smooth

_partition

of

_unity

we conclude that the

typical

sum to be

considered is the

_following

and

_w(h,

_d,

r,

s)

is the suitable smooth function

supported

in

with

_R,

VP.

In fact we _may

require

that S R since otherwise

by

the congruence

we would obtain

Therefore the

_change

of the roles of r and s involves the additional factor

e - h-’-

which contributes to the

_weighted

function

_{f (d, q, h, s, r).}

One

.

can

verify directly

that for 0

_v-

2,

i =

1, 2, 3, 4, 5

it holds

provided

which is satisfied whenever P

_{( ~ )2-E.}

We shall therefore assume it in the

sequel.

where

_f,

is the Fourier transform

_of/

_(taken

for ₇₇

_{= T~r’)}

divided

_by

Therefore

For k = 0 the Kloosterman sum

,S’(hdq",

s)

reduces to a

Ramanu-jan

sum for which it holds

hence the

_{corresponding}

contribution to

_R~P~(~,

_dq’,

_P)

does not exceed

which

_by

₍₉₎

is admissible since P

_{(.T, )2--,.}

For the

_remaining

_{range of 7~} we

apply

the smooth

partition

of

unity.

Then

_integrating

=

by

_parts

several times _we

away assume that k - li with

The sum we deal with has the

_following

form

where are

_arbitrary

complex

coefficients bounded

by

1 in absolue

value,

G is a function

satisfying

(13)

and the above sums is taken over

7~, ~

S.

To estimate it we shall

appeal

to the method of Deshouillers and Iwaniec

(see[2])

which

_provides

the estimates for multilinear forms in Kloosterman

6. Linear forms in Kloosterman sums.

LEMMA 2

_{~DESHOUILLERS, IWANIEC~.}

Let

_{D, H,}

S &#x3E; 1 and

_{~~d, Iz,}

_k,

_s)

be a smooth function

supported

in

~D, 2D~

x

~H, 2H~

x

~~f, 2I~~

x

~,S’, 2,5~

with

_partial

derivates

_satisfying

_(~3~.

Then for any

complex

numbers

bd,k

it holds

where

and

PROOF - See Theorem 11 of

_[2].

To

_simplify

the .C- term _suppose that and dominate in the

corresponding

expressions

in brackets. Then

Otherwise

Therefore in _any case

Let

In order to estimate

_P)

we distinct four cases :

I.

_{e=0, v=0.}

The

_{corresponding}

Kloosterman sum is

_equal

to

=

5’(hid,

kq;

s).

Hence

by

Lemma 2 and

separation

of variables

(see [2] p.269)

we have

with the variable li

_{replaced by KQ}

in the

_right-hand

side of

₍₁₅₎

thus

yielding

-where

Now we shall infer that in the

remaining

cases

R~P°~(x,

P)

admits better

estimates than in the above case.

The Kloosterman sum in

question

is

equal

to

S(hdq,

kq;

s)

=

S(hd,

k;

s)

The Kloosterman sum is

equal

to

S(hdq,

k;

s).

Hence

by

(16)-(18)

we obtain

The Kloosterman sum is

equal

to

S(hd,k,;s).

Since it

follows that

Therefore it is sufficient to consider the case

P).

Now we have

Hence for D = min

D2,

we obtain

7.

_Completion

of the

_proof.

Collecting together

the results of the

_previous

section we obtain

_by

₍₈₎

that

_5’(.c,P)

+

O(x~+‘

+

BX1-1!;)

provided

P

_(~)2-1!;

and D = min Since P _E

~2 x~, x~~

with

e

2 -

1 and B &#x3E;

_VQ

we obtain that the above conditions are satisfied

provided

,

Therefore the main contribution to

_{S(x) (see[l],[5])}

is to be evaluated on

the basis of

_equality

with

Hence

_{(cf. 1}

_defining

a =

f

we obtain that the total main term

log x ,

Therefore if

_1/2.

On the other hand

_by

an

obvious

_inequality

_{Px (~3, O) &#x3E; P~N- (,3,}

_8’)

whenever 8 ~

_0’,

we conclude

that

_{Px(,Q, O)}

&#x3E; xt. The

_proof

of the theorem is

_complete.

REFERENCES

[1]

J.-M. DESHOUILLERS and H. IWANIEC, On the _{greatest prime factor of n2} + 1,

Ann. Inst. _Fourier, Grenoble _32,

_4(1982),

1-11.

[2]

J.-M. DESHOUILLERS and H. IWANIEC, Kloosterman sums and Fourier coeffi-cients _of cusp forms, Inv. math. 70

(1982),

219-288.

[3]

C. _{HOOLEY, Applications of} sieve methods to the _{theory of numbers, Cambridge}

University Press, London

_(1976).

[4]

C. _HOOLEY, On the _{greatest prime factor of} a quadratic polynomial, Acta Math.

117

_(1967),

281-299.

[5]

H. IWANIEC, Rosser’s sieve, Acta Arith. 36

_(1980),

171-202.

[6]

H. J. S. _{SMITH, Report} on the _{theory of numbers,} Collected Mathematical _Papers, vol. _{I, reprint, Chelsea,} 1965.

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