### ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXX (1990) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

### Séria I: PRACE MATEMATYCZNE XXX (1990)

**J** **a c e k** ** P** **o m y k a l a** (Warszawa)

### Primes represented by a system of quadratic forms **1. Introduction. Statement of results. ** *Let P(xl5 x2, x3, x 4) and* *0(xl5 x 2, x3, x4) be quadratic forms with integer coefficients and let x be * a sufficiently large positive number. We deal with the following two problems.

### P roblem * 1. Find necessary and sufficient conditions for P(x) and (2(x) in * *order that there exist infinitely many primes p, q representable simultaneously by * *P and Q, respectively, i.e.*

**(1)** **(** **2** ^{)}

^{)}

*P(x) = p,* *Q(x) =q,*

*x e Z 4.*

**P** **r o b l e m** 2. *Assuming that the system * (l)-(2) *has arbitrarily large solutions * *in prime numbers p and q, estimate the quantity:*

### (3) *Г(х) = \{(p, q); p < x, q ^ x, p = P(x), q = Q{x), x e Z 4}|.*

### The natural, but not always necessary, assumption is the irreducibility of *P and Q. However, it is necessary that P(x) and Q(x) assume simultaneously * arbitrarily large positive values prime to any fixed natural number. This *condition will be automatically satisfied provided P, Q take simultaneously * positive values prime to a suitable integer depending on the coefficients of *P and Q only. Presumably the above conditions are also sufficient. This is * implied by the familiar H-hypothesis of Schinzel (see [11], [12]). Either problem in general seems to be very difficult. The special cases may be attacked by the Hardy-Littlewood method (cf. [15]) or the sieve method. Here we shall be concerned with the second one.

*We shall investigate Problems 1 and 2 for forms P, Q with separated * variables of the following type:

### (5) *P(x1, x2, x3, x4) = ÛUjpjfr!, x 2) + b<p2(x3, x4),* (6) Q(xl5 x 2, x 3, x 4) *= cq>1(x1, x 2) + d(p2{x3, x4),*

*where (p1{x1, x2) and (p2(x3, x4) are irreducible, integer-valued quadratic forms *

*of discriminants A1 and A2, respectively, with the same squarefree kernel*

**126** **J. P o m y k a l a**

*к = * *= к(Л2). The irreducibility of (pl and q>2 means that At (i = 1, 2)* *is not a square of an integer. By changing the coefficients a, b, c, d if necessary * *we can assume without loss of generality that the forms cpx and cp2 are * *primitive, and positive if definite. Let 3 — ad —be and let * *q* be the product of *all nonzero integers from the set {2, \a\, \b\, |c|, \d\, |<5|, \_AU d 2]} where *

*\_AX, A2] = l.c .m .^ , A2). Our main goal is the following*

**T** **h e o r e m** *1. The polynomials P(x), Q(x) defined by (5)-(6) represent simul*

*taneously infinitely many primes p, q if and only if there exist integers p0 > 0, * *q0 > 0, (p0q0, * *q* *) = 1, and an integer vector x° = (x?, x°, x°, x°) such that*

### (7) *P(x°) = Po,*

### (Ю *Q(x°) = q0.*

### If the equations (7) and (8) are solvable then there exists a positive *constant C depending on P and Q such that for sufficiently large x we have*

### (9) Г(х) ^ Cx(logx)""1.

### The proof of (9) will be divided into several parts according to the values *of 3, a, b, c, d. In the main case we follow the idea developed by H. Iwaniec in* [3], the ^-dimensional sieve being replaced here by the 1-dimensional one, since we consider the system of two equations. Also instead of Bombieri’s mean value theorem we are led to apply here a theorem of Barban-Davenport and Halberstam for estimating the error term.

### I would like to express here my gratitude to Professor H. Iwaniec for calling my attention to this problem and for stimulating discussions. I am also indebted to Professor A. Schinzel for the critical remarks concerning this paper.

### 2. *Case I (3 = 0). Throughout the paper we will regard p and q as prime * numbers. The relations (7) and (8) imply

### (10) ' *dp0 — bq0 = <5</>t (x?, x5) = 0, * *aq0- c p 0 = 3(р2(х°ъ, x2) = 0 * and the system ( 1 )— (2) is equivalent to one equation

*P(x) = p.*

*If ab = 0 then in view of (9), the answer to Problem 1 is provided by the * following

**L** **e m m a** *1 (see [10], [13]). Every primitive binary, irreducible, quadratic * *form, positive if definite, represents infinitely many primes.*

### For Problem 2, the following quantitative version of the above lemma is valid (see [14]):

### (11) *\{p ^ x; p = (p(xu x2)}| ~ Cvx/logx,*

*where Cv is a suitable positive constant, hence for Г(х) we even obtain the * asymptotic equality

*Г(х) ~ Cx/logx.*

**Primes represented by a system o f quadratic forms** **127**

**Primes represented by a system o f quadratic forms**

*If ab Ф 0, by (7) at least one of the forms acpl , bcp2 is positive definite if * *к < * 0 *, and (a, b) = 1. Let*

*(pl (x1, x 2) = A 1x\ + B 1x 1x 2 + Cl x* *2* *, * *(p2(x* *3* *’ ** 4 *) — A2x\ + В2х ъхА + C 2x\.*

### Replacing if necessary the forms *and tp2 by equivalent ones, we can assume * without loss of generality that

*aAx > 0, * *(Al ,b) = \, * *(A 2, a A 1) = \* *and — 4abA1A 2 is not a perfect square.*

### Considering the form

*(p{xx, x3) = aq)x{xx, 0) + b(p2(x3, 0) = aA1xj + bA2xl * we see by ( 1 1 ) that

*Г{х) ^ Cx/logx;*

### hence the estimate (9) in Case I is established.

**3. ** **Representation of integers by quadratic forms. Let <p be a primitive, ** *binary, positive if definite, quadratic form with discriminant A different from * *a perfect square. By Rv we denote the genus of q>. Define the following set of * primes:

**Representation of integers by quadratic forms. Let <p be a primitive,**

*M = <p;* = 1

**л**

### where *is the Kronecker symbol. For fixed a # 0 the symbol ^ | is * *a quadratic character with the conductor f(a), where*

### We have

*if k(a) = 1 (mod 4), * *if к (а) ф 1 (mod 4).*

### L emma 2 (see [3]). *I f a is different from a perfect square and * *denotes the* *( й \* *multiplicative group of residue classes prime to f (a) then the condition I — j = * 1 , *b'ef#, determines a subgroup * *of index 2 in 3 .*

*Using the convention a\bVjo(p\a=>p\b), let us recall that in [3] and [5] *

*(see also [4]) there has been constructed a certain set Г c {e > 0; e\d x ] and to * *each ее Г a certain set f£e of residue classes m od|d| has been found,*

### ( 12 ) *c <L (mod |zl|); (L, A) = 1,* ^{k(A)}

^{k(A)}

*L* *= 1*

### with the following property:

### (12') *(reZ, (r, A)= 1, L (mod\A\)e&e) =» r2L (m o d |d |) e ^ .*

**128** **J. P o m y k a l a**

**L** **emma** ** 3 ** (see **[3]). ** *An integer number n ф 0, positive if Л < * **0, ** *is represent*

*ed properly by one of the forms of genus R9 if and only if* *n = em, * *where е е Г , m (mod\A\)e£Fe, p\m=>pe&.*

**L** **emma** ** 4 (see [3]). ** *For any integer s Ф * **0 ** *there exists an integer r, * *(r, s) = 1, such that if n is represented properly by the genus R then r2n is * *represented by q>.*

**4. The main case (<5 ** **Ф** ** 0, ** **a b e d Ф** ** 0). ** First let us remark that without loss of generality we may assume in Theorem 1 that (x°, x®)^, (x®, x2)|p.

**Ф**

**a b e d Ф**

*To show this we find (by continuity of P(x)) a positive number a > * **0 ** such that

**P ( r ) > 0 , ** **<200 > 0 ** **(veK4)**

**whenever the distance ||c — x°|| is less than o. Next consider a sufficiently large ** **positive number X = A(x°, ** *q* *). Take any vector v e Z 4 satisfying*

**whenever the distance ||c — x°|| is less than o. Next consider a sufficiently large**

**positive number X = A(x°,**

*\\v —Xx°\\ < Xa/2, * *vt = xP(mod@), i = l , . . . , 4 .*

*By Dirichlet’s theorem for arithmetic progressions we find integers tt, * *i = 1, * 4, such that

### |tf| < Ясг(2^)-1 and *vf = Vi + tiQ = (xf, Q)pt,*

### where |p£|, i = 1, *4, are dinstinct prime numbers. Then the vector v°*

### satisfies the required conditions.

### The systems of equations (1)— (2) and (7)— (8) are equivalent to

### (13) *dp — bq = ôqyfx^ x2),*

### (14) *aq — cp = ôq>2(x3, x4),*

### (15) *dp0- b q 0 = ô(pfx0u x°2),*

### (16) *Щ0~ сРо = ô(p2{x°3, *2).*

*Setting rjt = (x?, x°) we see by (15) that the number* *n ! = (dp0- b q 0)/ôr]i*

*is a nonzero integer represented properly by q>x. Moreover, nx > 0 if A 1 < 0. *

### By Lemma 3, we have

### (17) *n1 = e1m1, * *et \A?, * *е1е Г 1, * (m1, d 1) = 1, *mx (mod\Ax\)e Jè?ei,* *Similarly, setting ц2 = (х2, x4) we may write*

### (18) *n2 = (aq0 — cp0)/ôr] \ = e2m2, where*

*e2\A2 , е2е Г 2, (m2, d 2) = l , m2 (m od|d2|) G ^ e2, p|m2= > pe^.*

**Primes represented by a system of quadratic forms** 129

**Primes represented by a system of quadratic forms**

*In view of Lemma 4 we find integers r1, r2 corresponding to cpx, <p2 such that * (19) *(r1 r2t Q ) = \ i * *(rx, r 2) = 1.*

*Define the number A = lt?0+ l el e2ôqjц\\ where 0 is the greatest exponent of * *the prime powers factorization of ([ml , m2], * *q* **°), i.e. 0 = max()i, for * (e00» Оч> w2]) = П rf1- We consider the system of congruences

*p = p0 (mod/l), * *q = q0 (mod^),* *dp — bq = r\ (modrf), * *aq — cp = r\ (modr2).*

*Since the moduli A, r\, r\ are pairwise coprime and {rxr2, abed) = 1, there * exists a solution *(mod/C), c2 (mod K) of the above system, where * *К = Arlr2 and (c1c2, K) = 1. We assume that*

### (20) *p = c1 (mod K), * *q = c2 (modX).*

*For such p, q we have*

*dp — bq = dp0 — bq0 = 0 (mod\ôrjlel rlge+1\); *

### hance for *= (m1, * *q* *m) we obtain* *dp — bq _ dp0 — bq0* ( 21 )

*à q \ e j x* *— * (modi?) *so that a, = - —z--- y—- is an integer, (a,, r,) = 1, satisfying*

*àn\exr \ f x*

### (22) /i<^i^i = (mod^)

### hence by (12), (12')

### (23) and *f lotl (mod |d 1|) G ^ 1.*

### An analogous argument shows that a2 *aq — cp*

*- — ---- -y-r is an integer satisfying* *àe2{q2r2Y f 2*

### (24) *(f2ct2, A2) = \ , * *f 2ot2 (mod |d 2|)e ^ fe2, * (a2, r2) = l . *

*We still have to determine the sector for p and q in which a t and a2 are positive * *if к < 0.*

### By (7H8), (15)— (16) the system of inequalities

*(dp0 — aq0)ô~1 > 0, (aq0- c p 0)ô~l > 0, p0 > 0, q0 > 0*

*determines the sector f f in which the point (p0, q0) is contained. Let us fix * *a square I in Sf with sides parallel to the coordinate axes such that * *(sl5 s2)e/=>0 < s1 ^ 1, 0 < s2 ^ 1. Our further considerations will be related * *to the homothetic square x l (x->oo).*

### Let

### (24') # = jp; ^{p} ^{XK, } ^{= } - l } .

^{p}

^{XK, }### Now we are in a position to prove

**— Commentationes Math. 30.1**

**130** **J. P o m y k a l a**

### L e m m a *5. Assume that * ( *p, q )exl satisfy (20). I f for z * > *i\a\ * + *\b\ * + |c| + *\d\)x112* (25) **(** **ol** **^** **p** **, q)x2(p, q),** П ^{p} ) = 1 >

**(**

**ol**

**^**

**p**

**, q)x2(p, q),**

*p < 2,р е&*

*then (dp—bq)ô~1 is represented by q>x and (aq — cp)ô~1 is represented by <p2.*

### P ro o f. First of all we observe that in view of (21), (al5 *q* ) = 1. Moreover, *(a1, r 1) = l. To show that (oc1, r 2) = l assume to the contrary that * *oi1 = 0 (modr) for some prime r |r 2. Hence by (20), dct —bc2 = 0 (modr) and * *ac2 — ccx = 0 (mod r). Since r f ô , we obtain r|c x, which contradicts the * *condition (cjc2, K) = 1. Therefore we have proved that (al , K ) = l . The * symmetry between af’s ensures that

### (25') *(oc1a2,K ) = l.*

*Now we prove that (dp — bq)S~1 is represented by q>1 (the case of cp2 being* *к*

**P** *f k \*

**P**

### (12), (17), (22) and (23) we obtain — = 1 , hence there must exist one more Va i /

### prime divisor p2lai such that — *l к*

**\P 2**

**\P 2**

*z2(\a\ + \b\ + \c\ + \d\)2 > a.l . This contradiction shows that p\ctl =>pe$; hence * *by (17), (23), (22) and Lemma 3 we find thatf 1ei a,1 is represented by R(pi and by * *Lemma 4, (dp — bq)/ôr]\ is represented by <pl5 i.e.*

*(<dp-bq)/S = * ^ x 2)

### for some (x1? x2) e Z 2. The proof of Lemma 5 is complete.

**5. The Rosser-Iwaniec sieve of dimension и е [ |, 2]. In this section we ** recall the estimates for the sifting function obtained in [7]. Following the standard notation (see e.g. [2]) we consider a finite sequence of positive integers

*, a2,...}*

*(afe ^ / means that at is an element of the sequence sé). For a given set 0 of * primes and z ^ 2 we write

**P(z) =** П **P-**

**P(z) =**

**P-**

*p e0 ‘, p < z*

### The main object in sieve theory is the sifting function *0 , z) which* *represents the number of elements a{Esé such that (a{, P(z)) = 1. For any * *m|P(z) we consider the subsequence stfm consisting of elements a{Esé such that * *at = 0 (modm). We assume that the number of а{Е ^ т is approximately equal * *to co(m)m~1X , where co(m) is a multiplicative function and X > 0 is a parame*

*ter (independent of m). Formally*

**(26) ** **KJ = — X+r(^,m),** _{m}

_{m}

*j = —1. But then p1p2\ot1 so oc1 ^ p^p2 ^*

*J = — 1. In view of Lemma 2,*

### similar). If *has a prime divisor p1 off 0 then*

**Primes represented by a system o f quadratic forms** **131**

**Primes represented by a system o f quadratic forms**

*where r( j/, m) is to be considered as a remainder term, X is to be chosen in * such a way that *, m) is small (at least on average). It is asserted that for the * *function co(m) and the set SP the following conditions are satisfied:*

### (27) *0 < œ(p) < p * *for peéP.*

*There exists a parameter x e [ j , 2 J such that*

### (28) Л / | о | Л ' { 1+ « L

*w i p < z \ * *P ) * *v o g w ) l * logw

*pe&>*

*for all z > w ^ 2, where L is a constant ^ 1, Щ = |$(w, z)| ^ 1. The parameter * *x satisfying (28) will be called the dimension of the sieve.*

### Let M > 1. For the sieve fuction we have the fundamental inequalities (see [7])

*m\P(z)*

### where Rosser’s weights A*(M) satisfy the conditions |2*| ^ 1, A* = 0 if *m > M. Using (26) we may write*

### + S ( r f , ^ z ) « ± ^ X ^{^m } ^{—} ^{ + } I *X i{M )r(d,m )*

^{^m }

^{—}

^{ + }

*m\P(z)* *m* _{m\P(z)}

_{m\P(z)}

### Let (29)

*± X G ±(M, * *z) + R ±(jrf, M) * (by definition).

*(o(p)'*

**v P ( z ) =**

### n f i

*pe&>*

*p < z*

*The following lemma provides an estimate for the main term G±(M, * *z).*

**L** **e m m a** ** 6 ** *(see [7]). Let s = logM/logz. Under (27)-(28) we have * *G + (M, 0>, z) ^ VP(z){Fx(s) + О( e ^ ~ s(logM)- 1 /3) * *if z ^ M ,* *G~(M, 0>, z) ^ Vp(z){fx(s) + 0(e'/ I - s(\ogM)-'L13) * *if z ^ M 1/p,*

*\2 * *if x = 1,* 1 *if x = i,*

*and the constant implied in the symbol O(-) may depend on x.*

*The only information we need about the functions Fx(s),fx(s), x = 1, 2 * or is given by the equalities:

*where*

*p* * = m =*

### (29')

*Fx(s) = Axs * * for s ^ 0 + 1 , s / *1 \ ~ xdt*

*f M(s) = Axxs * J ^ 1 - - J * *— * for 0 < s ^ 0 + 2,

*where A l/2, A t , A 2 are suitable positive constants.*

**132** **J. P o m y k a t a**

*Theorem 1 will be established in case II if we prove for stf = {a1a2; *

*p = cx (modX), q = c2 (modX), (p,q)exl} and 0* defined by (24') the * following

**T** **h e o r e m** ** 2. ** *For sufficiently small positive constant e * **> 0**

**> 0**

*(logxy*

*where the constant implied in the Vinogradov symbol > may depend on К but * *not on e.*

### In the first step we show the estimate

### (30) *S(s/, 0>, x ll2~e) > - --- -г£.* x2 *(log x y*

*In order to apply Lemma 6 we examine the quantity \s/m\, m\P(z). Setting * *F(Ci, C2) — (d^ — b£2)(a(2 —cCJ we find by (20) that for (m, К) = 1*

*W m\ = l{(p> q)exl: p = cx (modX), q = c2 (modX), a xa2 = 0 (modm)}|*

### I S ' .

*l^ Ç l,C 2 « m ;( Ç 1Ç2, X ) = l РЛ * *F(Ç*

*i*

*,Ç,2) = 0 (m o d m)*

*where the last sum is extended over (p,q)E xI such that*

### (31) *p = Cj (modX), Р = C\ (modm), q = c2 (modX), q = £2 (modm).*

*Furthermore, if (£., m) > 1 then there exists at most one prime p = £(. (mod m), * i = l , 2 , hence

### K J = s *X ' +0(Х/Ф(К)),*

### l ^ Ç b Ç *2 <m;(ÇiC2,m K ) = l P,q * *F(Ç i ,( 2) = 0 (mod m)*

*where ф(К) is Euler’s function and the constant implied in the symbol 0 ( ) is * *absolute. Let ht (mod mX) be the unique solution of the congruences * *ht = ct (modX), h( = £f (modm), / = 1 , 2 , such that*

### (ЗГ) *(hxh2, Xm)= 1.*

*Denote by \I\ the side length of I, and let (jq, y2) be the left-lower vertex of * *xl. Then the sum over p, q is equal to*

*{n(y1 + \I\x, Km, h1) — n(y1, Km, hi)}{n(y2 + \I\x, Km, h2) — n{y2, Km, h2)}.*

*Since for the function n(a, b, c) of primes p ^ a lying in the arithmetical * *progression c (mod b) we have the trivial estimate n(a, b, с) * *а/ф(Ь), we obtain* (32) ^ . {Ц0’. + 1 Л » ) - Ц у . у , + 1 Л х)-и уа} z ^{ф(т)_ 2}

^{ф(т)_ 2}

*Ф \ К ) * ÇbÇ2 (m odm )

*-\-r{sé, m).*

### Here and in the sequel, this last sum is to be understood as being taken over

*those ^ ^ ( m o d m ) for which (CiC2, m/X) = 1 and F(ÇX, £2) = 0 (modm),*

**Primes represented by a system of quadratic forms** **133**

**Primes represented by a system of quadratic forms**

### where and (33) provided

*i -ô* 5-0 о I

*li и = lim ( j (log t) 1 dt+ j log t) 1 dt)*

**1 + 5**

*r (s/,m )< * Z t 7 *ÿ u , , E{x, Km, hl) + ——*

*Çi,Ç2 (m odm ) Ф ( ^ ) Ф ( т ) * *Ф ( К )*

*E(a, b, c)* **7 r ( a , ** *b, c)-* *li a* *Ф(Ь)*

### For any prime r |K w e determine the number of solutions of the congruence *F((i, C2) = 0 (mod ra), a ^ 1, satisfying (£x£2, r) = L Let C2 = * *(Ç, r) = 1.*

### Then F(C1? C2) *= C iF(l, £). The quadratic congruence F(1, £) = 0 (mod ra) has * *exactly two solutions for rj(3. Since Ci runs over reduced residue classes, the * number in question equals 2</>(ra) = 2ra_1(r—1). The above argument shows *that the corresponding parameters X and œ are to be chosen as follows:*

### ( 34 ) *x = I11-Oh + 1Л *) —li Ti} {li (>’2 + |T|-x:) li y2} _ / * * V ^{2}

*ф(К)2 * *\(f){K)\ogx)*

*é(ra) * *2ra*

### (35) о И = 2 т Ц г ^ - — , *rJ(K.*

*ф(г*)2 * *Ф(гаУ*

*Now it remains to check that the condition (28) holds with x = 1. The Mertens * prime number theory provides the result

### L emma 7 (see [9]). *For 2 * ^ w < *z, (k , /) = * 1 *we have*

### 1 1 logz / 1

*- = T7^1° g 1--- + 0 . 1 -* *p * *ф{1) * log w

### I

**w $ p < z ** **p = k ( m o d i )**

**w $ p < z**

**p = k ( m o d i )**

*log W/’*

*where the constant implied in the symbol 0 Д ) depends on l. *

*Combining Lemmas 2 (à = k) and 7 we derive* *0>(p)\~l * *„ ( . * 2

**W ^ p < z** П

*pe&*

### l — *w ^ p < z \ * *n l 1 p - i* *P * *l* *р е »*

### = exP j — Z log ( 1 — 2 V *. w^p<z * *\ *

*ре»* P - 1

### = exp { Z *( ~+o( p* 2)

*p < z * *\P*

*р е »*

### = {exp _{t } **C ** _{\w^p<} ^{Z}

_{\w^p<}

### = е*Р1 Z ;

*, w ^ p < z P* *р е »*

### 1 + 0 w

**V****°****k**

### u ^{1 + 0} ^{w}

### I. logz

### = exp-{log777^+ 0 K logz

### log w

### log w 1 + 0 X

### 1

### log w 1 + 0

### w

### log w as required.

**134** **J. P o m y k a l a**

### Applying Lemma 6 for G~ with z = x1/2~£, M = x1_£, s = (logM)/(logz) ^

### ^ 2 + 2e we infer from (29') that the main term of the sifting function *0 , x 1/2_£) is equal to*

### (36) *XG~{M, 0 , x 1/2~E) = X7p(x1/2- 8){/1(s) + 0(e>/I" s(logM )"1/3)} *

### which is > £x2(logx)“ 3.

### In Section 7 we will prove the following bound for the remainder term:

### (37) *x 1_£) ^ Y* *j* * \r№ ’ m)l ^ -x:2/(logx)4*

*m< M*

### which together with (36) completes the proof of (30).

### The next task is to prove Theorem 2. Here the idea of changing the sieve dimension is applicable (see [3]). We consider the difference

*W (s/, e) = S { ^ , 0 , x 1/2~B) — S(jtf, 0 , x1/2+£)*

### and estimate the main constribution to it by means of the linear upper bound sieve. This will be done in the next section.

### 6. *Estimation of W{sé, £). Every element à e s é counted by the sifting * *function S(s/, 0 , x 1/2_£) but not by S(s/, * x1/2+£) has a prime divisor *p|P (x1/2+£) in the interval p e [ x 1/2_£, x1/2+e]. In virtue of (23), (24), à has an * *even number of prime divisors from 0 counted with multiplicity, so that there * *exists another p1|P(x1/2+£) (possibly p1 = p) such that p1 e [ x 1/2_£, x 1/2+£] and * *p p jd . Since pp1 ^ x 1~2e, we have the decomposition a = pp1a, with * *a\R?(x3E), where 0i1 = {pe J?; pJfK] and a |P f ( x 3£) means that a |P 1(x3£)00. *

### The number of such representations may be handled by adapting the linear *upper bound sieve for the sequence stfpa. The quantity W(s/, s) is then * estimated by the sum

### (38) I £ S ^ . ^ z , ) ,

*p e 0 > ,p < x 1/ 2 + e* a|RÎ°(x3e)

*where 0 >1 = (p; p e ^ u ^ , pJfKj, z 1 = x1/2_£, M = x1/2_5£. The main contri*

### bution to it amounts to the value

### z = £ £ *X{p,a)G+( M , ^ l , z l)*

*a \ R ? ( x 3e)*

*p < x l / 2 + c*

*with the corresponding parameters X , w to be chosen on the basis of the * equality

### l-^paml *X* *co(pam)*

*pam* *+ r(srf, pam).*

*We find X(p, a) = X* *co(p) cu(a) *

### p о and

### (39) ш(г) cu(ar) fco(r)/r *if 0 >1э г /)'а,*

*r * rco(a) (1/r *if 0 1эг\а,*

### the last equality being deduced from (35).

**Primes represented by a system o f quadratic forms** **135**

**Primes represented by a system o f quadratic forms**

### The estimate of the remainder term is postponed to the next section where it is proved that

### (40) JR(<s/, x 1-£) <| x2/(logx)4.

*Now we deal with the sum I . In view of Lemma 6 we have* (41) П ( l - — ' K F ^ H O ^ - O o g M ) - " 3)},

*p * *P * *a * *a * *p e & i * *\ * *P*

*p < z i*

*where s = log(x1/2_5e)/log(x1/2~8) and the summation over p is restricted by * *the condition pe0>, x 1/2~8 < p < x 1/2+E, while the inner sum is extended over * *a I R f ( x 3E). Therefore by (39) we obtain*

### (42) П *(1 -Ô (P )/P )=* П (1 — П I1 — 1/P)f

*p \ P l ( z i ) * *p \ P l ( z i ) * *p \ P i ( z \ ) * V *P * /

*p\a*

*=* ** î/p** 1 **( z i ) r i (** 1 **“** 1 **/p )(i-w (p )/p )_1.**

### Now the sum p|o

### 1 ° ^ ) П < 1 -> /р )(1 -® (р )/|’Г

### a 0 *p\a*

### may by expressed in terms of Euler’s product, namely

### I — П = П (1 *+ g ( p ) + g ( p 2) + . . . ) ,*

### a a *p\a * *p \ R ? ( x iE)*

*where the function g(n) is multiplicative, such that*

*g(p*)* = ^{C} *^ r - ^ - P ~ l) ( 1 - (°(P )P ~iy l* for a^ 1-

^{C}

### By (41) we obtain

### n (i+ 0 (p )+ 0 (p 2)+ ...).

*P * *P *

^{p}*\*

*r*

*T{.*

*x*

*3e)*

### It follows from (35) that *g(pa) * *= * *g(p)p1~a,* hence

### i *+g(p)+g(p2)+--- =* i + p ( p ) U + p _1+ p ~2+---}

### = i + 6f ( p ) ( i- p - 1r 1 = ^ i - *In view of (28) and Lemma 7 this yields (x = 2)*

### .3. V- <»(9)

*œ(p) -1*

*I <£ XV p^zJ log x'* *1*

### c 1/2-£</ *qet?* j < v 1/2 + i *q*

*(log x f* (logx) 2e2logx = 8"

### (logx) 3'

### 136 **J. P o m y k a l a**

### Concluding, by (38) and (40) we find that *W(s/, e) e2 x2/(logx)3.*

### Together with (36) the last estimate completes the proof of Theorem 2, provided that the estimates (37) and (40) are valid.

**7. The remainder term. Application of the Barban-Davenport-Halberstam ** **theorem.**

**T** **h e o r e m** *3 (Barban-Davenport-Halberstam). For any A > 0 there exists * *В — B(A) such that for x > x0(A) and Q ^ x/logBx the following estimate holds:*

*q* I I

*q ^ Q h = 1*

( M ) = 1
### The modified version of this theorem related to the well-known function *ф(х, q, h) instead of 7c(x, q, h) is proved in the book [8]. The above version is * obtained by applying the partial summation. In view of (37) and (40) we are led *to show that for M < x 1-£ one has*

*n(x, q, h) —* lix

*Ф(ч)*

*<i x 2 log Ax.*

### (43) *m^M Ui,Ç2(modm) Ф{К)(р(т) * Z i Z

^{, k ( k x}

^{> ( }

^{\ E}

^{( X ’ K}

^{m}

^{> h J}

^{+}

^{Â}*ф{К)J*

^{7}

^{ï \ \}### (logxf *(we recall that /г, = ^ (modm), hl = c l (mod/C)).*

*The contribution of the second term х/ф(К) is bounded trivially by * *Мхф(К)~1 <| x2-£/2 and hence may be ignored. The first term is estimated by* **(44) ** **X 2®"° I**

*M * *h (mod m) * *ф(К)ф(т)*

*(h,m) = 1*

*E(x, Km, h)*

*< </>(£)-1 (logM)( Z 2Q(m)/rn) £ E(x, Km, h).*

*m^M* *h = 1 *

*(h,m) — 1*

### By the Cauchy-Schwarz inequality, we get (45) *( X 2« “'/ш) X £ (*> Km>* * h)*

*M * *h = 1*

### = 1

### < { X *X E(x, Km, * *h)2Y l 2(* * X 4Q{m)/rny* ^{12}

^{12}*m ^ M * *h = 1 * *(h,m) = 1*

### m ^ M

### With the sum in the second bracket we deal as follows:

*X 4Q{m)/m ^ * X 1

*m $ M* *пцт2тзт4 ^ M 1* m, ... < I

*m < M* *m* ^ (lo g M + 1)4

**Primes represented by a system of quadratic forms** **137**

**Primes represented by a system of quadratic forms**

### On the other hand, in- view of Theorem 3 we have (cf. (43), (ЗГ))

**m ** *M K*

### (46) Z Z

^{E}

^{( x}

^{> K}

^{m}

^{’}

^{h ) 2}### ^ Z Z E(x» w> ^)2

**й = 1 ** *m ^ M K * *h — l*

**(h,m) — 1 ** **( f t , m K ) = l**

**(h,m) — 1**

### x2

### < (logx)14'

### Collecting (44)-(46) together we obtain (43) as claimed.

### 8. **Case III («5 ф 0, abed = 0). In view of symmetry we assume that a = 0. **

**Case III («5 ф 0, abed = 0). In view of symmetry we assume that a = 0.**

*Then be = — Ô ф 0. The equations ( 1)—* (2), (7}-(8) are reduced to (47) *p = b(p2(xz, x4), * *p0 = b(p2(x%, x£),*

### (48) *q = C(pl (xl , x 2) + ^p, * *q0 = etp^x0!, x%) + ^ p 0-,* *hence b = ± 1 and btp2 is positive if definite.*

*In case <7 = 0, by (48) we have c = ±1 and с(рх is positive if definite. *

### Theorem 1 follows by (11). Now till the end of the paper we can assume that *d ф 0, b = ± \ , b = \ if к < 0. The above equations are equivalent to*

### (49) *(q-bdp)c~l = <?!(*!, x2),*

### (50) *p = bcp2{x3, x4).*

### In view of (11) we may assume without loss of generality that

*\{p; p = b(p2(x3, x4), p = p0 (mode), (*3, * *xa* *) eZ 2 }\ > C ^x/logx* *for some positive constant C(f>2.*

### Following the notation of Section 4 we let *Â = * *\Qe+1el rl2lô\, * *K = Arl, * *and consider clf c2 (modK) satisfying the conditions: *

*cx = p0 (mod Â), c2 = q0 (modT), c2 — bdc1 = r\ (modr2), (c1c2, K) = 1, * and such that for the set

*Q = {p\ P = b(p2{x3, x4), p = c1 (mod KJ, (x3, x4) e Z 2}*

### we have

*\{p; p e Q , p ^ x}| ^ Ck x/logx.*

*Choosing a square Г with sides parallel to the coordinate axes such that * (sl5 s2)ef= > 0 < *^ 1, 0 < s2 < 1, (s2 — bdsl)c~l > 0, we define the sequence*

*stf = {a(p, q); p e Q , q = c2 (modK), (p, q )e x l’},*

*where a(p, q) = (q — bdp)/crile1f 1, with the aim of showing that for*

*^ ~ {p\ PXK, I - ) = - 1 } we have*

### (51) *S(stf, * x 1/2+£) x/(logx)5/2.

**138** **J. P o m y k a l a**

### For m|P(x1/2+£) we have

### (52) *K J = \{{p, q)exl'; p<=Q, q = c2(modK), q -b d p = 0 (modm)}|*

### = *Y * *Hl(<l)x(bdP) + 0 ( Y * !)>

*Z (mod m) p,q * *p,q*

*(pq,m)> 1*

*where x (mod m) is a Dirichlet character (mod m) and the sum Yp,q * extended *over (p, q)exl', p e Q , q = c2 (mod K).*

*The principal character Xo (mod m) provides the main contribution to *

### K J , namely the quantity

### (53) **ф(т** **) ~ 1 ** I 1 * = x ° ^ + 0 ( X * 1)

**ф(т**

**p>q ** **m ** **p,q**

**p>q**

**m**

**p,q**

### w i t h ( w . » ) = 1 ( № ” ) > !

*X = |{p; p e Q , p ^ х}\х/(ф(К)logx), * m(r) = ^ - y , r|P(x).

*Applying Lemma 6 (with к = \) we see that the main term is* *(54) G~{M, * x 1/2+£) x2/(logx)5/2 provided M ^ x“3, with a3 >

### Hence we now claim that

### (55) P K , M ) « x 2/( logx)3

*for M = x*3. We will show that (55) holds with M = x1_£. This is a conse*

### quence of the large sieve inequality and the Walfisz-Siegel estimate. In view of *(52), (53) the contribution of Yp,q>(pq,m)> i 1 to ^ K , M) is at most*

*Y (x1+£/2)/logx « x2_£/2;*

### hence it is negligible.

### We are led to estimate the quantity

### <£M_1 *Y \Lx(q)\\Lx(bdp)\,*

**X * X o q ** **P**

**X * X o q**

**P**

*where in the above sums p e Q runs over the interval p e [y 1? y l + x |/'|] and* (56) *q = c2 (mod K ), * *q e [y2, y2 + x |/'|],*

*{Ух->Уг) being now the left-lower vertex of xP. We replace the characters * *X (modm) by the corresponding primitive ones * *x** (modm*), m*|m, thus obtaining

### (57) *R ( ^ , M K Y * I

### ^ *Y Ф(т*1)~1 * *Y * *\ Z* *x* **(* *p* *) \ \ Y* *x* **(* *q* *)\-^-0{* *x* *2' bI2)*

*m*l ^ M * *x*(m odm *) p * *q*

### « lo g x *Y * *Ф(т) ~ 1* Z *\ Y x*( p) \\ Yx *{ q) \+ o( x 2' sl2).*

### Km^M *x*(m°dm*) p * *q*

**Primes represented by a system of quadratic forms** **139**

**Primes represented by a system of quadratic forms**

### Dividing the range 1 < m* ^ M into two intervals *'A = (1, (log*)5], * / 2 * = ((log*)5, x 1-*]*

### we find that

### (58) X l *r(s/,m )\< * X *ф{т*)~* 1 X *|X ^ *(‘?)| + 0 (*2~£/2).

*m e j i * l < m ^ ( l o g x ) 5 /* (m o d m * ) *q*

### Next, in view of (56) we have

### IXx*te)l = l X

**x * ( q ) \***q * **q = C2 (mod K)**

**q = C2 (mod K)**

*q ^ y 2 + x \ I ' \*

*= \ Ф ( к ) ~ 1 * X _ № 2 ) X *х * ( ч ) Ф ( я ) \*

*ф (mod K) * *q*

*q ^ y i + x \ r \*

### ^ I X *( х * Ф Ш \ -*

*q ^ y 2 + x \ I ' \*

*Since х * Ф Is a nonprincipal character (mod mK) one may apply the fol*

### lowing

### L e m m a *8 (Walfisz-Siegel, see [9]). For an arbitrary nonprincipal charac*

*ter x (modm), m ^ (logx)^ with N > 0, x ^ 3, there exists a constant * *C = C(N) such that*

### X *x ( p ) *

^{<}### * e x p (-C (N )4/iogx).

*p ^ x*

### The estimate

### X *(х*Ф)(ч) <x/(\ogx)4*

*q*Zy2 + x \ I ' \*

### combined with (58) shows that the corresponding contribution of m e / j to *M) is bounded by x2/(logx)3 as asserted.*

*It remains to deal with m e / 2. Dividing / 2 into intervals of the form * *{ J i , 2 Л ) (their number is ^ 21ogx) we reduce the problem to the estimate * (cf. (57))

**,S9) ** *X * *Ф(тГ' X \Х* *у* **{Р>\\Х* *у* **<Ч)\ <* ** A-** 2 **/(log.x)5.**

*J t ^ m < 2 J t * *p f m o d m ) p * *q*

### Its proof follows easily from

### L e m m a *9 (the large sieve inequality, see [8]). Let T{x*) = X« = * b + 1 * anX*in) * *with the primitive character x * (mod m). Then for any M ^ 1*

### m *B + N*

### Х т п ï |T(x*)i2 ^ ( M 2+7dV) X k l 2.

*m ^ M Ф { т ) y* ( m o d m ) * *n = B + 1*

**140** **J. P o m y k a l a**

### Applying to (59) the Cauchy-Schwarz inequality and Lemma 9 we derive

### Z < ^{p(m} r 1 Z |Zx*HIZx*M

^{p(m}*M ^ m < 2 . M * x * (m o d m ) p *q*

### Z WKm)-1 Z **\Zx*(p)\\Lx*(q)\**

**\Zx*(p)\\Lx*(q)\**

*, M ^ m < 2 M * x * (m odm ) p *q*

### Z ^{m0(m)_1 } Z Ег*И2}1/2

*, M ^ m < 2 J t * x * (m° d w ) p

### x { Z ^{гпф{ту} *1 Z * ^{E x * ( q} *)\2}112*

^{гпф{ту}

^{E x * ( q}*M ^ m < 2 M * *X* (mod m) q*

*<i J t* **~ 1 ** ^{{ ( J t 2} ** + x)x}1/2** ^{{ ( J t 2} ** + x)x}1/2 = ,,#_1(,/#2+ x)x **

^{{ ( J t 2}

^{{ ( J t 2}

**^ ** *J t x + x 2 J t ~ x* ** ^ x2(logx)-5 ,**

### hence in view of (54), (55), (59) we obtain (51). Thus case III is settled. The proof of Theorem 1 is complete.

**References**

**[1] H. D a v e n p o r t , Multiplicative Number Theory, Markham, Chicago 1967.**

**[1] H. D a v e n p o r t , Multiplicative Number Theory, Markham, Chicago 1967.**

**[2] H. H a lb e r s t a m and H. E. R ic h e r t, Sieve Methods, Academic Press, London 1974.**

**[2] H. H a lb e r s t a m and H. E. R ic h e r t, Sieve Methods, Academic Press, London 1974.**

**[3] H. I w a n ie c , Primes of the type <p(x, y) + A where (p is a quadratic form, Acta Arith. 21 (1972), ** **203-234.**

**[3] H. I w a n ie c , Primes of the type <p(x, y) + A where (p is a quadratic form, Acta Arith. 21 (1972),**

**[4] —, Primes represented by quadratic polynomials in two variables,"Bull. Acad. Polon. Sci. Sér. **

**[4] —, Primes represented by quadratic polynomials in two variables,"Bull. Acad. Polon. Sci. Sér.**

**Sci. Math. Astronom. Phys. 20 (3) (1972), 195-202.**

**[5] —, Primes represented by quadratic polynomials in two variables, Acta Arith. 24 (1974), ** **435-459.**

**[5] —, Primes represented by quadratic polynomials in two variables, Acta Arith. 24 (1974),**

**[6] —, On indefinite quadratic forms in four variables, ibid. 33 (1977), 209-229.**

**[6] —, On indefinite quadratic forms in four variables, ibid. 33 (1977), 209-229.**

**[7] —, Rosser's sieve, ibid. 36 (1980), 171-202.**

**[7] —, Rosser's sieve, ibid. 36 (1980), 171-202.**

**[8] H. L. M o n tg o m e r y , Topics in Multiplicative Number Theory, Springer, Berlin 1971.**

**[8] H. L. M o n tg o m e r y , Topics in Multiplicative Number Theory, Springer, Berlin 1971.**

**[9] K. P r a c h a r , Primzahlverteiluny, Springer, Berlin 1957.**

**[9] K. P r a c h a r , Primzahlverteiluny, Springer, Berlin 1957.**

**[10] E. S c h e r in g , Beweis des Dirichletschen Satzes, in: Gesammelte Werke, Bd. II, 1909, 357-365.**

**[10] E. S c h e r in g , Beweis des Dirichletschen Satzes, in: Gesammelte Werke, Bd. II, 1909, 357-365.**

**[11] A. S c h i n z e l, A remark on a paper o f Bateman and Horn, Math. Comp. 17 (1963), 445-447; **

**[11] A. S c h i n z e l, A remark on a paper o f Bateman and Horn, Math. Comp. 17 (1963), 445-447;**

**MR27, 3609.**

**[12] A. S c h in z e l and W. S ie r p in s k i, Sur certaines hypotheses concernant les nombres premiers, ** **Acta Arith. 4 (1958), 185-208; Corrigendum, ibid. 5 (1959), 259; MR 21,4936.**

**[12] A. S c h in z e l and W. S ie r p in s k i, Sur certaines hypotheses concernant les nombres premiers,**

**[13] H. W eb er, Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele ** **Primzahlen darzustellen fahig ist, Math. Ann. 20 (1882), 301-329.**

**[13] H. W eb er, Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele**

**Primzahlen darzustellen fahig ist, Math. Ann. 20 (1882), 301-329.**

**[14] Ch. d e la V a llé e P o u s s in , Recherches analytiques sur la théorie des nombres premiers, ** **Hayez, Bruxelles 1887.**

**[14] Ch. d e la V a llé e P o u s s in , Recherches analytiques sur la théorie des nombres premiers,**

**[15] A. I. V in o g r a d o v , On the Hardy-Littlewood binary problem, Acta Arith. 46 (1985), 33-35 ** **(in Russian).**

**[15] A. I. V in o g r a d o v , On the Hardy-Littlewood binary problem, Acta Arith. 46 (1985), 33-35**