On a theorem of Bombieri–Vinogradov typekowalski/fouvry-iwaniec-on-a-theorem.pdf · as a bilinear form (4), but unfortunately with L and M not well enough controlled for our method

MATHEMATIKAA JOURNAL OF PURE AND A P P L I E D M A T H E M A T I C S

VOL. 27 PART 2. December 1980 No. 54.

ON A THEOREM OF BOMBIERI-VINOGRADOV TYPE

E. FOUVRY AND H. IWANIEC

§1. Introduction. The celebrated theorem of Bombieri and A. I. Vinogradovstates that

, maxd/2)-e ( a , a ) = 1

n(x;q,a)-n(x)

4x(logxyA, (1)

for any e > 0 and A > 0, the implied constant in the symbol <g depending at moston E and A (see [1] and [14]). The original proofs of Bombieri and Vinogradovwere greatly simplified by P. X. Gallagher [4]. An elegant proof has been givenrecently by R. C. Vaughan [13]. For other references see H. L. Montgomery [10]and H. -E. Richert [12]. Estimates of type (1) are required in various applications ofsieve methods. Having this in mind distinct generalizations have been investigated(see for example [15] and [2]). Y. Motohashi established a general theorem which,roughly speaking, says that if (1) holds for two arithmetic functions then it also holdsfor their Dirichlet convolution; for precise assumptions and statement see [11]. Sofar, all methods depend on the large sieve inequality (see [10])

I I*q i Q *(mod q)

(2)

which sets the limit x1/2 for the modulus q in (1) and in its generalizations.It is the aim of this paper to present arguments which yield theorems of

Bombieri-Vinogradov type with an extended range for q. We shall treat carefully

n{x,z;q,a)= £ fz(n),n ^ x

n = a(modq)

where fz is the characteristic function of the set of integers n having no prime factorless than z. Let us introduce also

n{x ;z,q)= £ fz(n).

[MATHEMATIKA, 27 (1980), 135-152]

136 E. FOUVRY AND H. IWANIEC

We have proved the following

THEOREM. Let z < x1 / 8 8 3 and 1 < \a\ < x. Then, for any A > 0,

I n(x, z ; q, a)-—--- n(x, z ; q) (3)

the implied constant depending only on A.

Our method applies to a wide class of arithmetic functions f(n), for which the

sum

nf(x;q,a)= /(")n < x

n s u(mod q)

can be rearranged as a sum

lm = a (mod <y)

(4)

of bilinear forms, with the variables of summation / and m in appropriate intervals.Such a representation for fz(n) is obtained through a combinatorial sieve identity(see Lemma 1). We failed to obtain (3) in the most interesting case z = x1/2, in otherwords for f(n) = A(n). In the latter case, Vaughan's identity (see [13]) would serveas a bilinear form (4), but unfortunately with L and M not well enough controlled forour method to apply.

Acknowledgement. The authors express their gratitude to Professor H.Halberstam for pointing out some errors and for some helpful remarks about thefirst version of this paper. Our work was done when the second author enjoyed a oneyear visit to the University of Bordeaux I. It is a great pleasure for him to take thisopportunity to speak of the pleasing scientific atmosphere in which collaborationwas so fruitful.

§2. Sketch of the main ideas. The bilinear form (4) is approximated by

(lm,«) = 1

with the total error less than

R(M,L;Q)=Q<q<i2QM<m<i2M L < I sS 2L

(q,a) — 1 ("i,g) = 1 lm = a(modq)

a, -L < I si 2L

ON A THEOREM OF BOMBIERI-VINOGRADOV TYPE 137

The problem of bounding R(M, L ; Q) is reduced, by the Cauchy-Schwarz inequality

R(M, L;Q)^ {QM)ll2D1!2(M, L;Q),

to that of bounding the dispersion

( I I Y(q m \ L < I ^ 2L /

Im = a(modq) (/, q) = 1

= W(M,L;Q)-2V(M,L;Q)+U(M,L;Q),

say. Each of the terms U, V and W is evaluated separately, the most difficult beingW. By definition

W(M,L;Q) = E Z Z «/,<VQ < ^ 2 Q M < m ^ 2 M L < / i , / 2 ^ 2L

(y, a) = ! (m, q) = 1 l\m = /2m = a(mod^)

With an admissible error we may replace W(M, L; Q) by W*{M, L;Q), whichstands for the same sum with the range of summation restricted to (lx,l2) = 1>lt = /2(m°d <?)• In particular the diagonal terms lt = l2 disappear.

When treating W*(M, L; Q) we carry out the summation over m first. We use Jx

to denote the reciprocal of /j modulo g, so that Ij^ = \ mod<j. Writing

M

M < m « 2Mm = al, (modq)

it is trivial that \r(q, afjl ^ 1, but this turns out to be not satisfactory. We obtain agreat cancellation of the errors r(q, a l j in the sums over lt, l2 and q. By expandingeach r(q, al^ into a Fourier series, a typical term to be considered is

Wh(M,L;Q)=< , s ; 2 e L < / i , l 2 « 2 L

IJ,<I) = 1 / i s /2(modij)

with li f 0. Since Q will be nearly as large as L and /j = /2(mod q) there is not muchroom for summation over lt and l2. For this reason we reinterpret the condition/, = /2(mod q) by writing

h-l2 = qr with 0 < |r| < L/Q, (r, l,l2) = 1 .

Here r is r a t h e r smal l , so t h e c o n d i t i o n lx = / 2 ( m o d r ) c o n s t r a i n s t h e va r i ab l e s llf l2

less t h a n d o e s lt = I2(modq). In a d d i t i o n ,

r U


Therefore we arrive at sums of the type

ZZ0 < \r\ < L/Q

( l l , / 2 )

a,,a,2e( -ahr~),

with some Lt e (L, 2L], the factor

being removed by partial summation. A connection with the incompleteKloosterman sums is suggested. By the Cauchy-Schwarz inequality,

( l . / 2

Using Weil's estimate of XJ one just fails to get a non-trivial bound because themodulus /' /" is as large as the square of the length of the incomplete Kloostermansum Y,i- Hooley's conjecture R* (see [7]) would be helpful. In order to avoid anyunproved hypothesis, we appeal to a particular property of the coefficients a, torearrange the sum Xi,,/2

m t o a n° ther bilinear form with variables of summation of adifferent order of magnitude. Then, the above procedure yields incompleteKloosterman sums which are manageable by Weil's estimate. We doubt whether theelementary result of Kloosterman [9] is sufficient.

From the main terms in the dispersion D(M, L; Q) we get

M X <t>~2(i) ZQ < q ^ 2Q X (mod q)

101,-

L < I < 2L

We estimate this by applying the large sieve inequality and the Siegel-Walfisztheorem in a way familiar from the Barban and Davenport-Halberstam theorem.

§3. Lemmas. Let P{z) = H P < Z P for z $; 2. Let F(n) be an arithmetic functionvanishing for almost al t«. By the Buchstab identity

X fz(n)F{n) = X F(n) fP(n)F{n)) ,p<z \n = O(modp)

on applying the 'exclusion-inclusion principle' familiar from combinatorial sievetheory (see [5] and [8]) we obtain

LEMMA 1. Let D 7z z ^ 2. Then

E/,(n)F(«)=n 30(modd) X

d \ P(z)Z

nsO(modd)

ON A THEOREM OF BOMB1ERI-VINOGRADOV TYPE 139

where p(d) stands for the least prime factor of d, and, for a square-freed = pj ... pr > 1, px > ... > pr, we define

[ ( - I f , ifpl...Plpl<Dforalll^r,

{ 0, otherwise,and

( — l)r, if p^ ... p,p, < D for all I < r and px ... prpr ^ D ,

0, otherwise.

For d = 1 we define Aj = 1 and cr1 = 0.

Note that if ld j= 0 then d < D, and if ad + 0 thenobtain the

d < D. Hence we

COROLLARY. Lef D ^ z ^ 2. T/zew

Z /x(n)F(n + E Ep < z Dip1 Hd < D/p

pd | P(z)

X fp(l)F{dlp)

The following result is known in sieve theory as a 'fundamental lemma' (see [5]).

LEMMA 2. For R, z > 2 and (a, q) = 1 we /iat>e

E /.(«) = - n ( i - l N ) {n = a(mod(j) p|</

where s = logR/logz. T/ie implied constants are absolute.

LEMMA 3. If x(q) is the number of divisors of q,

nix 1(n,q) = 1

LEMMA 4. Let ip(^) = £, — [^]— ^ and A > 0. There are two functions A(£) andperiodic in £ (mod 1) such that

h + 0

B(h)e(hH) + A

withh ± 0

E. HJUVRY AND H. IWANIEC

and e(z) = e2nt2.

Proof. Take functions A(£) and B(£) of class C3 whose graphs are

n n+l*- A->

and whose derivatives satisfy Aip)(£), B(p)(£) <? A~" for p ^ 3, (compare with Lemma2 of [3]).

The next lemma is a consequence of Weil's estimate for Kloosterman's sums. Theproof is similar to Lemma 3 of Hooley [6].

LEMMA 5. Let 0 < A2-Ai ^ b. Then

A, <a a A2{a,be) = 1

a = A (mod A)

e(dr (b,d)ll2bll2T(bc)\og2b.

The implied constant is absolute. The notation a used when writing a/b or in acongruence (modi») means that ad s l(modfr).

LEMMA 6. For any pair a, b of non-zero coprime integers,

LEMMA 7. If x is a non-principal character modd, and d ^ ( l o g ^ , then

for any A > 0, the implied constant depending only on A.


Proof By Buchstab's identity,

E X(n)fz(n) = E Z(«)- E X(P) E X(n)fp{n)P < z n <i i

where zx = min(z,obtain

141

I X(n)fp(n)

. Letting R = £1/2 in the 'fundamental lemma' we

X(n)fP(n)n <i UP

= II (modi) P I <

PI i J

Pi

logi?

where sp = logR/logp ^ logK/logZj ^ ^(log^)1/2. The second double sum is emptyif z < exp (yAog £); thus we assume that z > exp (>/log <) = zx, and we obtain

E X(n)fp(n)= E Z(«) Ei H/p n < {/zi zi < p =S rain (z,{/n,p(n))

« £ nn s£ £/zi '*

by the Siegel-Walfisz theorem. This completes the proof.

COROLLARY. Under the same assumptions,

E y(n) fJn) <

( n , e ) = 1

§4. Reduction of the problem. We split up the sum (3) into < (log x)2 sums oftype

E /Z(«)-TT3 E /,(«)S(y, Q)= EQ < < ? « ; _ .

(q,a) = 1 n = a(modq) (n,q) - 1

with 2y sg x and 2g ^ x11/21. It is sufficient to show that

S(y,Q) <x(\ogx)-A-2,

for z s£ x1/883 and 1 ^ \a\ ^ x. We have trivially that

(5)

E. FOUVRY AND H. IWANIEC

so that (5) is obvious for y < x(logx) ^"2. In what follows we assume that

x(logx)- < y ^ x. (6)

Now we want to rearrange S(y, Q) as a sum of bilinear forms. For this, apply Lemma1 twice to the characteristic function of the set of integers n e (y, 2y], n = a(modq)and to the characteristic function of the set of integers n e (y, 2y], {n, q) = 1. Thensubtract l/<t>{q) times the second inequality from the first, to obtain, as in thecorollary to Lemma 1,

Z L(n) -y <n ^ ly

n = a(mod</)

L(n)y <n ^ 2y(n,i}) = 1

I 1-Id < D y < n ^ 2yd\ PU) " = "(mod^)

(d,q) = 1 n = O(modrf)

y yn = O(modd)

+ z z fPin) ~p<z Dp~2 ^ d < Dp~^ y < n ^ 2y

pd \ P{z) n = a(modq)(pd,q) = 1 n = O(moddp)

y<n^2y "n = 0 (mod pd)

() i

p <zpit

say. Hence, in the above notation

Sp(q,D)< z g < <J s: 2(2

() 1

(7)

say. The sums Sp(y,D,Q) with p < min(z, exp(%/logx)) = z0, say, will beestimated easily by means of Lemmas 2 and 3 while those with larger p will betreated by a dispersion method.

§5. Estimate o/S^j;, D, g). For (d, q) = 1 we have

Z

and, by Lemma 3,

y < n <i lyn = a(modg)n = O(modii)

Zy<n(,ly

n = O(modii)


Hence S^q,/)) -4 D and consequently

Sl{y,D,Q)<QD<xi~u, (8)

provided

QD^x 1 " 2 8 , (9)

which we henceforth assume.

§6. Estimate of^,p < z0Sp(y, ^> 2)- By Lemma 2, for each a with (a, q) = 1, wehave

V /•(«) = — ITpmn s a (mod^) pi | q

where i? is any number ^ p and sp = logi?/logp. Hence

Dp-i < m a DP-I \pqm j q p

and consequently

P P

For /? = xE/2 this bound yields

^ Sp(y, D,Q)<x exp ( - (log x ) 1 / 3 ) , (10)P < Z0

the implied constant depending on e only.

Now we proceed to estimate Sp(y, D, Q) with z0 ^ p < z.

§7. Rearrangement of Sp{y,D,Q). Let M take the values 2"'Dp~1 forA = 1, 2,... such that Dp~2 < M < Dp"1, so that there are at most 21ogp such M's.We split up Sp(y, D, Q) into < logp sums of the type

EJy, M,Q)= ^ ^ ^ ' - 1

thus obtaining

< pmn ^ 2y ^rXH/ y < pmn ^ 2j{q, ap) = 1 (m, q) = 1 pmn = tj (mod g) (n, pg) = 1

(P, n) = 1

, B, Q) X ^ P ^ ' M - 2)+ 0 (wr 2 log x). (11)M


Here, the error term comes from the contribution of rc's divisible by p2. This error isadmissible because

X yp ~2 log x < y(log x) exp( - ,/log^x).zo sg p < z

§8. The dispersion. By the Cauchy-Schwarz inequality we obtain

E2p(y,M,Q)^MQDp(M,Q),

where Dp(M, Q), called the dispersion, stands for

DP(M,Q)= X I ( I / P W - ^ ^ /p(n)T(ij,ap) = 1 (m, q) = 1 pmn = a{modq) {n,pq) = 1

= Wp(M,Q)~2Vp(M,Q)+Up(M,Q),

say. Each term Up, Vp and Wp will be evaluated separately. By definition,

UP(M,Q)= £ X ( J _ S /p(n)YQ < i } 5 ; 2 g M < m « 2M vPVH) y < pmn 2y(g,aj>) = 1 (m, g) = 1 (n.P«) = 1

§9. Evaluation of Vp(M, Q). By definition,

^ /p(»l)/p(»2)g < q s: 2Q V l y J JV/2 < ni ,n2 < 2JV Mi < m < M2f(j,ap) = 1 {n\n2,pq) = 1 m apn\{modq)

1/2 < ni/«2 < 2

where for simplicity we have written JV = y/pM,M1 = max(M,^/(p«1), y/(pn2))and M2 = min(2M,2y/(pn1),2y/(pn2)). We carry out the summation over m first.Trivially we would take {M2-Mi)q~1 + 0(1) but this is useless because M is goingto be smaller than Q. Therefore we are looking for an explicit formula for the errorterms, with the expectation of obtaining substantial cancellations when summingthem over q. We begin with

M\ <_m ^ M2m — apn\ (modq)

1 = l ^ j - ^ l l + ^ _ i "^-M _ ,/, "'•» - i ^ i I (12)

where \j/(d) = 0 - [ 0 ] - j . To arrive at UP(M, N) we replace (Mj-MJq" 1 , with thehelp of Lemma 3, by

, 2 \<l>(q)J

The first term above contributes to Vp(M, Q) exactly UP(M, Q), while the error term


0{x{q)l4>{q)) contributes

„ . x(q)< 2 N —2— ^ N Q log Q .

Therefore we may write

VP(M,Q) = Up(M, Q)+ Vp(Mt, Q)- Vp{M2, Q) + O(N2Q~1 logQ).

where for L = M1 or M2 we define

145

(13)

Q < q s: 2Qiq,ap) = 1

N/2 < m,n2 < 2JV(mttj.pq) = 1

1/2 < ni/«2 < 2

apni

By Lemma 4 we approximate tp(£) by A(£) with error B(^) giving

* ± 0 AT/2 < n i , n 2 < 2N QqHQ(q,apn\n2) = 1

• (14)

We use v | q°° to mean that each prime that divides v also divides q. Sinceq/<t>(q) = Xv|«=oV^1' w e obtain, by Lemmas 5 and 6 and by partial summation,

i ê^-^y s / Lpn1-a\ ( q \e h e ah I

Q q Q{q, apn\rt2) —

supPQNJ Q < e, «2g 7

q, apnin2) = 1

l\ an

sup V " 1

(v,dpn\n2) = 1 (r, apn\r\2) = 1

\h\x

Now, summing over n1,n2<2N and h ^ 0 with weight Cfc, (14) withA = MQ~rx 2c yields


Hence relation (13) becomes

Vp(M, Q) = Up(M, Q)

Q-1x5e). (15)

§10. Rearrangement of Wp{M, Q). By definition,

Wp(M,Q)= XQ<q H2Q M < m « 2M (nin2,p) = 1{q, ap) = 1 (m, q) = 1 yjpm < n\, ni ^ 2y/pm

pmni = pmti2 = a{modq)

Here, if (nt, n2) = d > 1 then d > p. Therefore, such terms contribute at most

O{MN2Q'lp-1+MNlogx). (16)

Now, let us consider W*{M, Q)—the contribution to WP(M, Q) of terms with(n1; n2) = 1. Notice that the range for nl, n2 is equal to

<%(q) = {(ni,n2);(n1n2,pq) = l . K , . ^ ) = 1,

JV nnx = n 2 ( m o d q ) , — < n1, n2 < 2N ,j < — < 2 } ,

2 n2

a n d for given q,n1,n2 t h e n u m b e r of m's is given by (12) wi th the same n o t a t i o n forMt and M2. We treat the main term (M2 — Mi)/q as in Section 9. On replacing it by

y iM, < m « M2

we make the total error ^ N2Q~X logQ. Another error of order (16) is made whenrelaxing the condition {nx, n2) = 1 in S(q). The latter operation is necessary toobtain

TP(M,Q)= X J - X X (G q « 1Q r W ) ((mod^) M < m « 2M \ )> < pmn « 2y{q,ap) = 1 (/, q) = 1 (m,q( = 1 pmn = t{modq)

() 1

which we consider as a main term for Wp(M, Q). It is clear, by the above discussion,that we obtain

Wp(M,Q) = Tp(M,Q) + 2Wp(M1,Q)-2Wp(M2,Q)


where for L = Mt or M2 we define

WP(L,Q)=Q <q sj 2Q (BI,B2)E[q, ap) = 1 BI > i

§11. Rearrangement of Wp(L,Q). Now approximate i ( ) by /l(^), with error£(£), and expand -4(<!;) and £(£) into Fourier series (see Lemma 4) giving

Wp(L, Q) < AN2\ogN+ f CJW;>fc(L, g) | , (17)h= 1

where

?, ap) = 1 ni > "2

Replace the condition (q,p) = 1 by splitting up the summation over q into p —1arithmetic progressions a(modp), 1 ^ a < p, and detect (q, a) = 1 by the relation

f 1, if (g,a) = 1,

v q \ 0, it (g,a) > 1,

to obtain

wP.k(L'Q)= I ZMvTO(L,g), (18)1 ^ a < p v | a

with

/ P K ) / > 2 ) e ( ^ apn2

g s a(modp) n\ > 112q = 0{modv)

For (nj.njje^q), we reinterpret the condition ny = n2(modq), by writing«i-«2 = qr> so that 1 < r < NQ'\Qr < n^-n2 < 2Qr,

nt—n2 = ar(modpr), n1—n2 = O(modvr) and (n1n2,pr) = 1 .

We may therefore write

1 «r«Af/e(ni,B2)6if(r)


where q = (nl —n2)/r and the range of summation in the inner sum is

N; y < n1?n2 < 2N,n2 < n, < 2n2,

Qr<n1-n2^2Qr,(n1,n2) = l,

{nx n2 •> prv) = 1, nl — n2 = otr(mod pr),

«! = n2 (mod vr) >.

The variables n2 and n1 are of the same order of magnitude. Our intention is to spoilthis 'symmetry' by an appeal to the combinatorial sieve identity. We apply Lemma 1,with some parameter G > p in place of D, giving

Ig < G (n2,prv) = 1 nj = O(modî)

+ X IG (02,^^) = 1

say.

+ zG/p !g 9 sc G (12

(20)

§12. Estimate of

obtain, by Lemma 6,

z . Since g = a(modp) and q =. rnj(modn2) we

L-apn2 Lpn2-a q r(Lpn2-a)= 1- a = 1- a

q qpn2 pn2 {nl-n2)pn2 pn2(modi).

Insert this into the inner sum £B1 and remove the factor e(hr(Lpn2 — a)lpn2(nl —n2))


by partial summation, to obtain

149

1=pn2

hxS U P

JV/2 < iVi < 2iVPn2

by Lemma 5. This yields

I Ig ^ G n2

§13. Estimate of £ X

(mod WJ, Lemma 6 yields

(21)

. Since g = a(modp) and q = — rn2

{

Insert this into the inner sum £ n i / a n d remove the factor e(hr(Lpnl —a)/pn1(nl — n2))by partial summation, to obtain

« (i+J1^-) sup yPNQJ N/2 < N, < 2JV JV/2 < n2 < JV 11 < Wl

(^•prv) = 1 MI = O(modg)

an,where c(nx) =/p(9)(n1)e( aî—-J is independent of n2. By the Cauchy-Schwarz

inequality,

N/2 < n2 < JV ni ^ Ni(n2,prv) = 1 » | s OJmodg)

1/2

n', n " ^ JVi (ri, n) e .^n' = n" = O(modj) (n", n) e J»(r\

(n'n", p) = 1 I

HP-

1/2

However,

np n —ri np


Hence, by Lemma 5,

X e (ahr ( ~ - ^ ]) < (ah{n"-n'), rin"fl2(n', i(n. .(«", n) e J^fr)

Summation over «' and n" yields

n\ n" < 2Nri = n" ~ 0(mod3)

h<h< 2N/g

Gathering all the above results together we finally obtain

I XG/p < g < G n2

(22) 1

§14. Estimate of Wp(L, Q). Collecting (17), (18), (19), (20), (21) and (22) weobtain

Wp(L,Q) ( ^ ^

- 1 x 7 e , (23)

for A = MQ~lx~lt and G = (a ,p)"2 / 5p"1 / 5N2 / 5 .

§15. Estimate of the dispersion Dp(M, Q). If we introduce results (15) and (23)into the definition of DP(M, Q) we obtain

Dp(M, Q) = Xp(M, Q) + RP(M, Q),

where XP(M, Q) stands for the sum of the main terms, i.e.

Xp(M, Q) = Tp(M, Q)-2Up(M, Q)+Up(M, Q)

e < 9 < 2 e M < m « 2 M VV</^ l(modg) \ y < pmn < 2y lT\H> y < pmn « 2y(q,ap) = l ( m , i } | = l (/,()) = 1 pmn = /(mod?) ( n , p ? ) = l

(n,P)= 1

and the error term Rp(M, Q) is

^ ( a , ? ) 1 / 1 0 ? 1 3 ' 1 0 ^ 2 9 / 1 0 ^ - ^ 7 8 . !


It remains to estimate Xp(M, Q). For this, we appeal to the large sieve inequality (2).We first write

Z ( Z / , ( » > - T ^ Z/{modq) \ y < pmn ^ 2y{/, q) = 1 pmn = l(modq)

4>(<i) y < pmn « 2 / y < pmn ^ 2y(n,pq) = 1

Summing this over <j and replacing each x (mod q) by its induced primitive characterX*(modd), d q, we obtain

1

G < « = ; 20 >• < pmn

<Q-2(\ogQ)2

• < 2Q 1 < d < 2Q/e x (mod d) y < pmn ^ 2y(n,ep) = 1

If Q/e ^ (logx)2/1+11 = /J, say, apply the Corollary to Lemma 7, and if Q/e > Happly the large sieve inequality (2), to show that this expression is

whence

and finally

DP(M, Q) 4

Xp(M,Q) <

(24)

§16. Conclusion. By (11),

P <

zo « p < z

152 ON A THEOREM OF BOMBIERI-VINOGRADOV TYPE

on taking D = x[0/2[-i£ for Q sj x11'21 and z ^ x1/883. Hence by (7), (8) and (10) we

obtain (5). This completes the proof of the theorem.

Remark. The limit for the present method turns out to be Q = x10119^'1 in whichcase z ^ xd, with 5 = d(n) a very small positive constant.

References

1. E. Bombieri. "On the large sieve", Mathematika, 12 (1965), 201-225.2. X. Ding and C. D. Pan. "A new mean value theorem", Sri. Sinica, Special Issue (II), (1979), 149-161.3. J. Friedlander and H. Iwaniec. "Quadratic polynomials and quadratic forms", Ada Math., 141 (1978),

1-15.4. P. X. Gallagher. "Bombieri's mean value theorem", Mathematika, 15 (1968), 1-6.5. H. Halberstam and H. -E. Richert. Sieve Methods, (Academic Press, London-New York, 1974).6. C. Hooley. "On the number of divisors of quadratic polynomials", Ada Math., 110 (1963), 97-114.7. C. Hooley. "On the greatest prime factor of a cubic polynomial", J. Reine angew. Math., 303/304

(1978), 21-50.8. H. Iwaniec. "Rosser's sieve", Ada Arith., 36 (1978), 171-202.9. H. D. Kloosterman. "On the representation of numbers in the form ax2 + by1 + cz1 + dt2", Ada Math.,

49 (1926), 407^(64.10. H. L. Montgomery. Topics in Multiplicative Number Theory, Lecture Notes in Math. 227 (Berlin and

New York, 1971).11. Y. Motohashi. "An induction principle for the generalization of Bombieri's Prime Number Theorem",

Proc. Japan Acad., 52 (1976), 273-275.12. H. -E. Richert. Lectures on Sieve Methods, (Tata Inst. of Fund. Research, 1976).13. R. C. Vaughan. "On the estimation of trigonometric sums over primes and related questions", Institut

Mittag-Uffler Report No. 9 (1977).14. A. I. Vinogradov. "The density hypothesis for Dirichlet's L-series" (Russian), Izv. Akad. Nauk SSSR,

Ser. Math. 29 (1965), 903-934; Corrigendum: ibid. 30 (1966), 719-720.15. D. Wolke. "Ober die mittlere Verteilung der Werte zahlentheoretischen Funktionen auf Restklassen I,

Math. Annul, 202 (1973), 1-25.

Dr. E. Fouvry, 10H15: NUMBER THEORY; Multiplicative theory;U.E.R. de Mathematiques et d'Informatique, Distribution of primes and of integers withUniversite de Bordeaux I, specified multiplicative properties.351, Cours de la Liberation,33405 Talence,France.

Dr. H. Iwaniec,U.E.R. de Mathematiques et d'Informatique,Universite de Bordeaux I,351, Cours de la Liberation,33405 Talence,France. Received on the 9th of April, 1980.

Documents

On a theorem of Bombieri–Vinogradov typekowalski/fouvry-iwaniec-on-a-theorem.pdf · as a bilinear form (4), but unfortunately with L and M not well enough controlled for our method