(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)
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
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:
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)
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.
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
\\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
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 } .
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 >
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 \
(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
z2(\a\ + \b\ + \c\ + \d\)2 > a.l . This contradiction shows that p\ctl =>pe$; hence by (17), (23), (22) and Lemma 3 we find thatf 1ei a,1 is represented by R(pi and by Lemma 4, (dp — bq)/ôr]\ is represented by <pl5 i.e.
(<dp-bq)/S = ^ x 2)
for some (x1? x2) e Z 2. The proof of Lemma 5 is complete.
5. The Rosser-Iwaniec sieve of dimension и е [ |, 2]. In this section we recall the estimates for the sifting function obtained in [7]. Following the standard notation (see e.g. [2]) we consider a finite sequence of positive integers
, a2,...}
(afe ^ / means that at is an element of the sequence sé). For a given set 0 of primes and z ^ 2 we write
P(z) = П P-
p e0 ‘, p < z
The main object in sieve theory is the sifting function 0 , z) which represents the number of elements a{Esé such that (a{, P(z)) = 1. For any m|P(z) we consider the subsequence stfm consisting of elements a{Esé such that at = 0 (modm). We assume that the number of а{Е ^ т is approximately equal to co(m)m~1X , where co(m) is a multiplicative function and X > 0 is a parame
ter (independent of m). Formally
(26) KJ = — X+r(^,m), m
j = —1. But then p1p2\ot1 so oc1 ^ p^p2 ^
J = — 1. In view of Lemma 2,
similar). If has a prime divisor p1 off 0 then
Primes represented by a system o f quadratic forms 131
where r( j/, m) is to be considered as a remainder term, X is to be chosen in such a way that , m) is small (at least on average). It is asserted that for the function co(m) and the set SP the following conditions are satisfied:
(27) 0 < œ(p) < p for peéP.
There exists a parameter x e [ j , 2 J such that
(28) Л / | о | Л ' { 1+ « L
w i p < z \ P ) v o g w ) l logw
pe&>
for all z > w ^ 2, where L is a constant ^ 1, Щ = |$(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\P(z) m 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
(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 \ 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
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 )
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
= 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|Rΰ(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
![(x) denote the number of cube-full numbers in the interval [1, x], where x is a sufficiently large number. On the one hand, an asymptotic formula for Q](https://data11.123dok.com/thumbv2/9liborg/000/798/798302/cover.webp)
![A major branch of combinatorial set theory concerns set mappings, i.e., functions f : X → P (X), where one seeks for a large free subset , that is, a subset Y ⊆ X such that x 6∈ f (y) for any two distinct x, y ∈ Y . P. Erd˝ os [2]](https://data11.123dok.com/thumbv2/9liborg/000/801/801017/cover.webp)
![Suppose x is a real, x# exists and P ∈ L[x] is a poset on ω1](https://data11.123dok.com/thumbv2/9liborg/000/801/801835/cover.webp)
![Abstract. It is proved that for any Banach space X property (β) defined by Rolewicz in [22] implies that both X and X](https://data11.123dok.com/thumbv2/9liborg/000/796/796704/cover.webp)
