Primes in tuples II. Cem Yalçin Yıldırım. Bo gaziçi University Istanbul, Turkey. 1. Introduction. p n+1 p n log p n. = lim inf

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1 Acta Math., , 47 DOI: 0.007/s c 200 by Institut Mittag-Leffler. All rights reserved Primes in tuples II by Daniel A. Goldston San José State University San José, CA, U.S.A. János Pintz Hungarian Academy of Sciences Budapest, Hungary Cem Yalçin Yıldırım Bo gaziçi University Istanbul, Turkey. Introduction In the first paper in this series [7] we proved that, letting p n denote the nth prime, = lim inf n! p n+ p n log p n = 0,. culminating 80 years of work on this problem. Since the average spacing p n+ p n in the sequence of primes is asymptotically log p n, this result showed for the first time that the prime numbers do not eventually become isolated from each other, in the sense that there will always be pairs of primes closer than any fraction of the average spacing. For the history of this problem, we refer the reader to [7] and [8]. The information about primes used to obtain. is contained in the Bombieri Vinogradov theorem. Let θn; q, a = n N n a mod q { log n, if n is prime, θn, where θn = 0, otherwise..2 The Bombieri Vinogradov theorem states that for any A>0 there is a B =BA such that, for Q=N /2 log N B, max N θn; q, a φq a,q= q Q N log N A..3 The first author was supported in part by an NSF Grant, the second author by OTKA grants No. K 67676, T 43623, T and the Balaton program, the third author by TÜBITAK.

2 2 d. a. goldston, j. pintz and c. y. yildirim Thus the primes tend to be equally distributed among the arithmetic progressions modulo q that allow primes, and this holds for the primes up to N and at least for almost all the progressions with modulus q up to nearly N /2. The principle can be quantified by saying that the primes have an admissible level of distribution ϑ or satisfy a level of distribution ϑ if.3 holds for any A>0 and any ε>0 with Q = N ϑ ε..4 Elliott and Halberstam [3] conjectured that the primes have the maximal admissible level of distribution, while by the Bombieri Vinogradov theorem we have immediately that 2 is an admissible level of distribution for the primes. In [7] we proved that if the primes satisfy a level of distribution ϑ> 2 then there is an absolute constant Mϑ for which p n+ p n Mϑ for infinitely many n..5 In particular, assuming the Elliott Halberstam conjecture or just ϑ 0.98 gives p n+ p n 6 for infinitely many n..6 These are surprising results because they show that going beyond an admissible level of distribution 2 implies that there are infinitely often bounded gaps between primes, and therefore questions as hard as the twin prime conjecture can nearly be dealt with using this type of information. Since we obtained our results in 2005 there has been no further progress toward.5, and it appears now that an extension of the Bombieri Vinogradov theorem of sufficient strength to obtain bounded gaps between primes will require some basic new ideas. One can also pursue improving the approximations we used or improving the method used to detect primes, and again this now appears to require some essentially new idea. Our goal in this paper is to extend the current method as much as possible in order to obtain strong quantitative results. In particular, we obtain the following quantitative version of.. Theorem. The differences of consecutive primes satisfy lim inf n! p n+ p n <..7 log pn log log p n 2 This result is remarkable in that it shows that there exist pairs of primes nearly within the square root of the average spacing. By comparison, the best result for large gaps between primes [7] is that lim sup n! p n+ p n log p n log log p n log log log p n 2 log log log log p n 2eγ,.8

3 primes in tuples ii 3 where γ is Euler s constant. Thus the best large gap result produces gaps larger than the average by a factor a bit smaller than log log p n, while now the small gaps are smaller than the average by a factor a bit bigger than log p n /2. In this sense the small gap result now greatly surpasses the large gap result. There are conflicting conjectures on how large the gap between consecutive primes can get, but all of these conjectures suggest that there can be gaps at least as large as clog p n 2 for some constant c. This paper is organized as follows. In 2 we present a generalization of Theorem which applies to many interesting situations and which is the result that we will prove in this paper. Unlike in [7], to obtain our results we need to take into account the possibility of exceptional characters associated with Landau Siegel zeros. In Theorem 2 we assume that there are no Landau Siegel zeros in a certain range and are able to obtain our main results including the gaps between primes in Theorem in intervals [N, 2N] for all sufficiently large N. Next, using the Landau Page theorem, we can find a sequence of ranges which avoid possible Landau Siegel zeros. Thus we obtain Theorem 3 which unconditionally gives the same results as Theorem 2 but without being able to localize them to a dyadic interval. The proof of these theorems requires substantial refinements of the methods of [7], and in 3 we will discuss some of these refinements and how they arise. The main technical tools needed in our proof, Theorems 4 and 5, are stated in 4. The proofs of Theorems 4 and 5 take up 5 3. In proving Theorem 5 in our general setting, we need a modified Bombieri Vinogradov theorem which is the topic of 2. Our method, as in [7], requires a result on the average of the singular series. In [7] the well-known result of Gallagher [5] was used, but in our current setting this result is not applicable, and therefore in 4 we prove a new result well adapted for our needs. With Theorems 4 and 5 in hand together with the new singular series average result, the proofs of Theorems 2 and 3 are completed in 5. Notation In the following c and C will denote sufficiently small and sufficiently large absolute positive constants, respectively, which have been chosen appropriately. This is also true for constants formed from c or C with subscripts or accents. We will allow these constants to be different at different occurences. Constants implied by pure o, O and symbols will be absolute, unless otherwise stated. The ν times iterated logarithm will be denoted by log ν N. P denotes the set of primes.

4 4 d. a. goldston, j. pintz and c. y. yildirim Acknowledgment We thank the referee who was helpful with many corrections and improvements in this paper. 2. A generalization of Theorem Our method will allow us to prove a generalization of Theorem where instead of seeking two neighboring primes of the form n+j and n+j with j < j h, h = hn = C log n log log n 2, 2. we look for two primes of the form n+a j and n+a j, where n [N +, 2N] and A = {a j } h j= [, N] 2.2 is an arbitrarily given set of integers, such that 2. is satisfied. We remark that an extension of this type is a trivial consequence of the prime number theorem if h = h n > +c log n, c > 0 fixed, 2.3 but that none of the earlier methods of Erdős [4], Bombieri Davenport [] and Maier [3] which produce small gaps between primes seem capable of proving a result of this type for any function satisfying h = h n < c log n, c > 0 fixed. 2.4 According to a conjecture of de Polignac [9] from 849, every even number may be written as the difference of two primes. Although we know that this is true for almost all even numbers, there is no known way to specify these values. Our generalization makes a first step in this direction by proving that we can explicitly find sparse sequences A such that infinitely many of the elements in A A are differences of two primes, i.e. P P A A =. 2.5 Here we make use of the usual notation A B={a b:a A and b B} for sets A and B. Some sequences for which our method applies are: A = {k m } m=, k 2 fixed k N, 2.6 A = {k x2 +y 2 } x,y=, k 2 fixed k N, 2.7 A = {k fx,y } r=, k 2 fixed k N, 2.8

5 primes in tuples ii 5 where the value set R={r N:fx, y=r for some x and y} satisfies RX = {m R : m X} > C X log 2 X 2.9 this would happen, for example, for fx, y=x 2 +y m fx, y=x 3 +y 3, or in general any set of type with arbitrary m 2, and for A = {k rj } j=, k 2 fixed k N, 2.0 if R={r j } j= N satisfies the density condition 2.9. Among these sets the only trivial one is 2 m, that is 2.6 for k=2 which has log X/log 2 elements below X, thereby corresponding to the case 2.3. Unfortunately, the possible existence of Landau Siegel zeros cf. 2 makes it impossible to formulate a localized version of our result for all n N satisfying the conditions 2. and 2.2. Even in the special case when A is an interval, we cannot guarantee the existence of gaps of size O log N log 2 2 N between primes in any interval of type [N, 2N] for any large N. The first formulation of our main result assumes that there are no exceptional characters in a certain range. This is hypothesis SY from 2, with Y =Y N=e 3 log N. Theorem 2. Let us suppose that an N >N 0 is given such that for any real primitive character χ mod q, q e 3 log N, we have ] Ls, χ 0 for s 9 log N,. 2. Let A=A N ={a j } h j= [, N] N be arbitrary a j a j if j j with h C log N log 2 2 N, 2.2 where C is an appropriate absolute constant. Then there exists n [N, 2N] such that at least two numbers of the form are primes. n+a j and n+a j, j < j h, 2.3 In 2 we show that 2., i.e. conjecture SY N, is true for an infinite sequence N =N ν!, and thus Theorem 2 implies Theorem by choosing N =N ν and A = A N = {, 2,..., h} 2.4 with h= C log N log 2 2 N. In a similar way, Theorem 2 implies the following result too. Theorem 3. Let A N be an arbitrary sequence satisfying AN = {n A : n N} > C log N log 2 2 N for N > N Then infinitely many elements of A A can be written as the difference of two primes, that is, P P A A =. 2.6

6 6 d. a. goldston, j. pintz and c. y. yildirim 3. Some initial considerations The main tool of our method is an approximation for prime tuples and almost prime tuples. Consider the tuple n+h, n+h 2,..., n+h K as n runs over the integers. If these K values are all primes for some n then we call this a prime tuple, and we wish to examine the existence of prime tuples. A first consideration is that the set of shifts H = {h, h 2,..., h K }, with h j h j if j j, 3. imposes divisibility conditions on the components of the tuple which can effect the likelihood of obtaining prime tuples or even preclude the possibility of more than a single prime tuple. Specifically, let ν p H denote the number of distinct residue classes modulo p occupied by the elements of H, and for squarefree integers d extend this definition to ν d H multiplicatively. The singular series for the set H is defined to be SH = p K ν ph. 3.2 p p If SH 0 then H is called admissible. Thus H is admissible if and only if ν p H<p for all p, while if ν p H=p then one component of the tuple is always divisible by p and there can be at most one prime tuple of this form. Hardy and Littlewood [9] conjectured an asymptotic formula for the number of prime tuples n+h, n+h 2,..., n+h K, with n N, as N!. Letting { log n, if n is prime, θn = 0, otherwise, 3.3 we define Λn; H := θn+h θn+h 2... θn+h K 3.4 and use this function to detect prime tuples. The Hardy Littlewood prime-tuple conjecture is the asymptotic formula Λn; H = NSH+o, as N!, 3.5 n N which is trivial if H is not admissible, but is otherwise only known to be true in the case K =, which is the prime number theorem. The starting point for our method in [7] is to find approximations of Λn; H for which we can obtain asymptotic formulas similar to 3.5. A further essential idea is that rather than approximating just prime tuples, we should approximate almost prime

7 primes in tuples ii 7 K-tuples with a total of K +l prime factors in all the components, which, in case 0 l K 2, guarantees at least two of the components being prime. The almost prime tuple approximation used in [7] and which we also use here is Λ R n; H, l := K +l! d P H n d R µd log R K+l, 3.6 d where H =K and P H n := n+h n+h 2... n+h K. 3.7 Our method for proving. in [7] is based on a comparison of the two sums Λ R n; H, l 2 and θn+h 0 Λ R n; H, l n N n N An asymptotic formula for the first sum can be obtained if R N /2 ε, while for the second sum we can use an admissible level of distribution of primes ϑ to obtain an asymptotic formula when R N ϑ/2 ε. In [7] it was assumed that K and l are fixed, i.e. independent of N. Using these asymptotic formulas we can now evaluate S R := 2N n=n+ h 0 h θn+h 0 log 3N h,h 2,...,h K h distinct Λ R n; H, l If S R >0 then the sum over h 0 must have at least two non-zero terms and thus there must be some n and h j h j such that n+h j and n+h j are both prime. We find with ϑ= 2 and h=λ log N for any fixed λ>0 that we can choose K and l for which S R>0, which proves.. In order to obtain this for any arbitrarily small λ>0, the fixed K and l are chosen sufficiently large in an appropriate way. To obtain quantitative bounds to replace., the first step is to obtain asymptotic formulas which are uniform in K and l so that these can be chosen as functions of N that go to infinity with N. One also needs explicit error terms, and these error terms arise not only from lower order terms and prime number theorem type error terms, but also in 3.8 from the Bombieri Vinogradov theorem error terms. We now establish the relations between our parameters that will be used throughout the paper. Recalling the set A from 2.2, we will always take H A. Next, R and l will be chosen as K h, l K and R = 3N /4 o. 3.0 We will make use of two important parameters U and V defined by V := log N and U = e V, 3.

8 8 d. a. goldston, j. pintz and c. y. yildirim and will choose K later to be slightly smaller than V. We next denote the product of primes not exceeding V by P := p, 3.2 p V where p will always denote primes. As just mentioned above, our present treatment requires a much more delicate analysis of the error terms than in [7], and therefore we make an initial simplification to facilitate this analysis. In [7] the irregular behavior of ν p H for small primes greatly complicated the estimate of the function Gs, s 2 and its partial derivatives. We can avoid these difficulties, at least for primes dividing P, by proceeding somewhat similarly to Heath-Brown in []. We call a residue class a mod P regular with respect to H and P if P, P H a = 3.3 and denote by AH=A P H the set of all regular residue classes mod P. Thus AH := {a : a P and P, P H a = }. 3.4 The number of regular residue classes mod P is clearly AH = p Pp ν p H 3.5 and their proportion of all the residue classes mod P is AH P = p P ν ph, 3.6 p which is positive if H is admissible. Thus in particular for a given H and all P there exists at least one regular residue class mod P, if and only if H is admissible. With this notation, we now consider the sums 2N n=n+ n AH AH 2 Λ R n; H, lλ R n; H 2, l 3.7 and 2N n=n+ n AH AH 2 Λ R n; H, lλ R n; H 2, lθn+h 0, 3.8

9 primes in tuples ii 9 with h 0 [, h], which are asymptotically evaluated in Theorems 4 and 5, respectively. A new feature in the proof of these theorems which does not occur in [7] is that 3.7 and 3.8 are first evaluated for each residue class a mod P with a AH AH 2 = AH H separately, and then the results are added over all regular residue classes mod P. It turns out that the asymptotic main term and even secondary terms are independent of the particular choice of the regular residue class a, so this summing presents no difficulty. However, the restriction of the values of n to a single residue class a mod P in 3.8 requires a stronger form of the Bombieri Vinogradov theorem cf. 2. where To detect primes, in place of 3.9 we consider S RN, K, l, P := Nh 2K+ 2N n=n+ Ψ RK, l, n, h := p:p n A H: H =K n AH log p log 3N Ψ RK, l, n, h 2, 3.20 Λ R n; H, l. 3.2 On applying Theorems 4 and 5 we can asymptotically evaluate S R, which we carry out in 5. One condition that arises from the main terms is that, in order to prove the existence of prime pairs in intervals of length h, we need Since K h, this immediately implies that h > C log N K h > C log N Our goal is to take h as small as possible, and therefore we cannot obtain anything better than 3.23 when using the approximation in 3.6 together with Apart from powers of log 2 N, we are able to prove our results for h of this size. Our actual choices for K and h are K c log N log 2 2 N and h = 25 log N K 25 c log N log 2 2 N, 3.24 with a sufficiently small explicitly calculable absolute constant c to be chosen later. We will need the error terms in Theorems 4 and 5 to be uniform in K with a relative error of size η satisfying η < c K. 3.25

10 0 d. a. goldston, j. pintz and c. y. yildirim However, we do not achieve this for all admissible pairs H and H 2 of size K. Instead, for all admissible pairs H, H 2 we obtain a weaker error term, but if K H H 2 K 3.26 then we do obtain the error estimate in This turns out to be sufficient for our proof, since such pairs H, H 2 will be dominant in Two basic theorems In the following, let N be a sufficiently large integer, c a sufficiently small positive constant, K c log N log 2 2 N, 4. K k, k 2 K and K l, l 2 K. 4.2 We will consider sets H:=H H 2, H, H 2 [, N], of sizes Let H = k, H 2 = k 2 and H H 2 = r. 4.3 m := K m for m [0, K], n := max{ K, n} and n := n. 4.4 Our first main result is the following theorem. Theorem 4. For N c <R N /2 e c log N we have, as N!, Λ R n; H, l Λ R n; H 2, l 2 n N n AH AH 2 l +l 2 log R r+l +l 2 = N r+l +l 2! l +O Ne c log N. SHP +O AH K r log 2 N log R 4.5 For the next theorem we suppose that the following instance of the Bombieri Vinogradov theorem holds see 2. For a given, sufficiently large N, and recalling the parameter P defined in 3.2, we have max log p N a,q= ϕp q N P e c log N, 4.6 where q Q q,p = N<p 2N p a mod P q with an arbitrary positive constant c. Q = N /2 P 3 e c log N, 4.7 Letting H 0 =H {h 0 }, our second main result is as follows.

11 primes in tuples ii Theorem 5. Suppose that N satisfies 4.6 and 4.7, and let N c R Q. Then Λ R n; H, l Λ R n; H 2, l 2 θn+h 0 N<n 2N n AH AH 2 = N C Rl, l 2, H, H 2, h 0 l +l 2 SH 0 log R r+l+l2 r+l +l 2! l K r log +O 2 N +O Ne c log N/log 2 N, log R where, if h 0 / H, l +l 2 + log R C R l, l 2, H, H 2, h 0 = l +r+l +l 2 +, if h 0 H \H 2, l +l 2 +2l +l 2 + log R l +l 2 +r+l +l 2 +, if h 0 H H 2. For the applications to Theorems 3, the simpler case l =l 2 =l will be sufficient Lemmas We will use standard properties of the Riemann zeta function ζs. Proceeding slightly differently from [7], we use the zero-free region, with s=σ+it, { } ζ+s 0 for s R N := s : σ. 5. log 2 N +6 log t +3 Further, by Titchmarsh [20, Chapter 3], for s R N we have { max ζ+s s, ζ+s, ζ ζ +s+ } s log t In the course of the proof the following contours which lie in the zero-free region R N will be used with U and V given in 3.: { L := s : σ = } { and t U, L 2 := s : σ = } and t 2U, 28V 4V { L 3 := s : σ = } { and t U, L 4 := s : σ = } and t 2U, V 4V { L 5 := s : σ } { and t = U, L 6 := s : σ } and t = 2U, 28V 4V and where L = L 0 L, 5.4 L 0 = { s = δ 0 e iϕ : 2 π ϕ 3 2 π} and L = {s = it : δ 0 t U}, with δ 0 = K log 2 N.

12 2 d. a. goldston, j. pintz and c. y. yildirim Similarly to Lemma of [7], we have the following. Lemma. Let k log 2 2 N log R, N c R N, N C, k 2 and B Ck. Then log t +3 B R s ds L j s k e c log N, 3 j 6, 5.5 where the constant implied by the symbol depends only on C. Proof. The integral I in 5.5 satisfies, for all j, 2U I R /28V 0 e c C log N 0 log t +3 B max{ t, /28V } k dt+ 28V k dt+ C dt t 3/2 dσ R σ σ /4V U 3/2 +e log N/4 /2 e c log N. We will prove a generalization of the combinatorial identity 8.6 of [7] in order to evaluate the terms I, of 8. Let us define for triplets of integers d, u, y with d 0, u 0 and y+u 0 to be called suitable triplets the quantity Zd, u, y := u! u m=0 m y u m dd+... d+m. 5.6 m y+m! Lemma 2. We have for any suitable triplet d, u, y the relation Zd, u, y = y d+... y d+u. 5.7 u!y+u! Proof. We will prove this by induction on u. For u=0 we have trivially, for any non-negative d and y, Zd, 0, y=/y! the empty product in the numerator of Z is by definition. We can suppose that u and that our statement is true for all suitable triplets d, u, y. We set, for any real number x and n<0, x n! = in other words, we just neglect in a sum all terms with an n! in the denominator with n<0 and fix the notation [S] = {, if the statement S is true, 0, if the statement S is false.

13 primes in tuples ii 3 Then we obtain by u j = u j u Zd, u, y = u! function j=0 j y u u j + u j u j=0 j y where we define u u = u j dd+... d+j y+j! u j dd+... d+j j y+j+! =0, = u j dd+... d+j d+j u! j y+j! y+j+ j=0 j y u [ y 0] y dd+... d y 2d y y = u u j dd+... d+j y+ d u u! j y+j+! j=0 j y = y+ d Zd, u, y+ u y+ dy+2 d... y+u d =. u!y+u! Finally we mention a simple lemma for the mean value of the generalized divisor d m q := m ωq, 5.9 where ωq denotes the number of prime factors of q for a squarefree q. Lemma 3. If m>0 and ν max{c logk +, }, then there exists a constant C depending on c such that, for K and x we have and q x q x d m q x+log x m 5.0 d 3K q +/ν +log x C K, 5. q where the means that the summation runs over squarefree values of the variable. Proof. Equation 5.0 follows from q x d m q x q x Further, by 5.9, we have d m q q x j x m x+log x m. 5.2 j d 3K q +/ν = d j q 5.3

14 4 d. a. goldston, j. pintz and c. y. yildirim with j = 3K +/ν 9e /c K, 5.4 and Lemma 3 follows with C =9e /c Preparation for the proof of Theorem 4 Since the preparation for the proof of Theorem 4 is nearly the same as in 6 and 7 of [7] for the analogous Proposition or 3 or 4, we will briefly summarize it and the reader is referred for the details to [7]. Let Hp = {h,..., h ν : ph h j h g mod p, h g H for some g, h j p}, 6. and ν p H H 2 := ν p H p H 2 p = ν p H +ν p H 2 ν p H. 6.2 For any a AH AH 2 cf. 3.4, we have, similarly to 7 of [7], S R N; H, H 2, l, l 2, P, a := 2N n=n+ n a mod P Λ R n; H, l Λ R n; H 2, l = N P T Rl, l 2, H, H 2 +OR 2 3 log R 7K, where, denoting by the integration over the vertical line {s:re s=}, T R l, l 2, H, H 2 := R s R s2 2πi 2 F s, s 2 s K+l+ s K+l2+ ds ds 2, F s, s 2 := ν ph p ν ph 2 +s p + ν ph H 2 ; 6.5 +s2 p +s+s2 p>v here, differently from [7], primes not exceeding V do not appear in F s, s 2 since, by the regularity of a, P H n, P =P H2 n, P =. Let := j<j K Then, if p consequently, for all sufficiently large primes p, h j h j N KK / ν p H = H = K, ν p H 2 = H 2 = K and ν p H H 2 = H H 2 = r. 6.7 We therefore factor out the dominant zeta-factors and write ζ+s +s 2 d F s, s 2 = G H,H 2 s, s 2 ζ+s a ζ+s 2 b 6.8

15 primes in tuples ii 5 with a function Gs, s 2, regular for σ j > 5, say, which we write slightly more generally for future application in Theorem 5 as G H,H 2 s, s 2 = Gs, s 2 = G = G G 2 G 3 = G G 4, 6.9 where now a=b=k, d=r, ν p=ν p H, ν 2 p=ν p H 2, ν 3 p= ν p H H 2, G s, s 2 = p V p +s a p V p +s2 b p V p +s+s2 d, 6.0 and G 4 s, s 2 = p>v = p p>v ν p p ν 2p +s p + +s2 p +s ν 3p p +s+s2 a p +s2 =: G 2 s, s 2 G 3 s, s 2. p p>v Let us use the notation b p +s+s2 d 6. and δ j := max{0, σ j }, j =, 2, δ := δ +δ 2, s 3 := s +s 2, s 3 := s +s { R N := s : σ 2 log 2 N +6 log t +3 }. 6.3 We will estimate the order of Gs, s 2 in the region s, s 2 R N under the more general conditions a, b, d K, ν j p K, j =, 2, 6.4 ν p = a, ν 2 p = b and ν 3 p = d for p. 6.5 Later we will examine more delicate properties of Gs, s 2 with further conditions on a, b, d, ν p and ν 2 p. We have G s, s 2 exp C K p δ p V G 2 s, s 2 exp C K p δ p G 3 s, s 2 exp C p>v K 2 p 2 2δ e CK log 3 N, 6.6 exp CK p δ e CK log 3 N, p log +o 6.7 e CK2 /V e CK, 6.8

16 6 d. a. goldston, j. pintz and c. y. yildirim where we made use of the estimates max{v δ, log δ } log 2 N /2 log 2 N = e. 6.9 Further, in 6.7, the sum which was originally over p has been majorized by using the set of the smallest possible primes which could divide. Summarizing , we obtain and further Gs, s 2 e CK log 3 N for s, s 2 R N, 6.20 F s, s 2 e CK log 3 N log t +3 log t K for s, s 2, s 3 R N. 6.2 The above estimate shows that the integrand in 6.4 vanishes as either t! or t 2!, s, s 2, s 3 R N. We will examine the integral analogously to 7.5 in [7] I := TRd, a, b, u, v, H, H 2 := Ds, s 2 R s+s2 2πi 2 s u+ s v+ 2 s +s 2 ds d ds 2, 6.22 where we introduce the function Ds, s 2, regular for s, s 2, s 3 R N : where Ds, s 2 := D 0 s, s 2 Gs, s 2, D 0 s, s 2 := W d s +s 2 W a s W b s 2, W s := sζ+s, d a, b K, min{a, b} ck, 8 K u, v K 6.24 and by symmetry we may assume that u v. In the applications we will have a b and u v. First step. Move the contour for the integral over s to L, and the one over s 2 to L 2. The vertical parts t U and t 2U, respectively, can be neglected similarly to Lemma. After this, move the integral over s from L to L 3 L 5. The horizontal segments L 5 can again be neglected. We pass a pole of order u+ at s =0, and obtain I = I +I 2 +O e c log N, 6.25 where I := 2πi = 2πi L 2 u! L 2 Res s=0 u j=0 Ds, s 2 R s+s2 u j ds 2 s +s 2 d 2 Ds, s 2 R s log R s +s 2 s=0 d s v+ ds 2 2 s u+ s v+ u j j s j

17 primes in tuples ii 7 and I 2 := 2πi 2 L 2 L 3 Ds, s 2 R s+s2 s u+ s v+ 2 s +s 2 ds d ds We denote the complete integrand above by Zs 2 and express j s j Ds, s 2 s +s 2 s=0 d = j D0, s 2dd+... d+j s d+j 2 j j j + j Ds, s 2 j = s j s=0 dd+... j j d+j j, s d+j j where, for j=j including also the case when j=j =0 and d 0 arbitrary the empty product in the numerator is. Second step. Let us denote the contribution of the first term in 6.28 to 6.26 by I j, 0 and the others by I j, j, j j. Then I j, 0 belongs to the main term, while all of the I j, j with j just contribute to the secondary terms. Let us move now the contour L 2 for the integral over s 2 to L 4 L 6 in The horizontal segments L 6 can be neglected again. We pass a pole of order v++d+j j in case of I j, j and we obtain in this way I = u! u j=0 u log R u j j v+d+j j + 2πi ν=0 j j dd+... d+j j j j j v+d+j j! j =0 v+d+j j log R v+d+j j ν ν ν Zs 2 ds 2 +O e c log N L 4 =: I, +I,2 +O e c log N. s ν 2 j s j Ds, s 2 s=s 2= Estimates of the partial derivatives of Ds, s 2 In this section we will estimate the partial derivatives j s j j s j 2 Ds, s 2 of Ds, s 2 for j+j CK with s j =s j in R N estimate for functions regular in z z 0 η: for j 3. We will often use Cauchy s j! f j z 0 η j max fz. 7. z z 0 =η

18 8 d. a. goldston, j. pintz and c. y. yildirim Applying this to Ds, s 2, we obtain j!j! j s j j s j 2 Ds, s 2 η j+j max Ds, s s s η s 2 s 2 η In order to substitute the above maximum for Ds, s 2, we have to estimate { } Ls, s 2 := max log Ds, s 2 s, log Ds, s 2 s for s, s 2, s 3 R N ; since by the regularity of log Ds, s 2 for s j R N, j 3 cf. 6.0, 6. and 6.23, we have for η 00 log 2 N +log t +3+log t 2 +3, Ds, s 2 Ds, s 2 2η exp max Ls, s s s η s 2 s 2 η By symmetry, it is enough to deal with L s, s 2 := log Ds, s 2 s. 7.5 Since the logarithm is an additive function, using the representation and 6.23 of Ds, s 2, it is sufficient to examine the factors D 0, G, G 2 and G 3 separately. We will choose a positive η such that η 00 log 2 N +log T, T = T +T 2, T j = t j +3, j =, 2, 7.6 where, by δ=δ +δ 2 2/log 2 N, V =log N /2, log K 2 log N log 2 N, we have max{v δ, V η, log δ, log η }. 7.7 We have, by 5.2 and 6.23, ζ log D 0 s, s 2 = d s ζ +s +s 2 + ζ a s +s 2 ζ +s + K log T. 7.8 s Further we have s log G s, s 2 = p V log p p +s dp s2 p +s+s2 a p +s K log 2 N. 7.9

19 primes in tuples ii 9 Similarly to 6.7, we obtain by 7.7, log G 2 s, s 2 s = log p ν p ν 3 pp s2 p +s ν pp +s ν 2 pp +s2 +ν 3 pp +s+s2 p p>v a p + dp s2 +s p +s+s2 K log p p δ p K p log +o K log 2 K log 2 N. log p p δ Finally, analogously to 6.8, we have by 6.5, s log G 3 s, s 2 = p p>v p>v K 2 p>v log p p +s log p a K p δ p log p p 2 2δ a dp s2 ap +s bp +s2 +dp +s+s2 δ +dpδ2 K a p + +s p δ dp s2 p +s+s K2 V 2δ K2 V K. Summarizing , we have max Ds, s 2 e CηKlog 2 N+log T Ds, s 2, if s, s 2, s s s η 3 R N. 7.2 s 2 s 2 η estimate. Hence, 7.2 and 7.2 imply by the choice /η=00klog 2 N +log T the following

20 20 d. a. goldston, j. pintz and c. y. yildirim Lemma 4. We have for s, s 2, s 3 R N, j j j!j! s j Ds, s 2 s CKlog j 2 N +log T j+j Ds, s The above estimate is sufficient for our purposes at every point s, s 2 apart from 0, 0, which will appear in the main term. We will show an analogous result for the point 0, 0 where η in 7.6 will be replaced by the larger value η 0 = d log 2 N, 7.4 where we use the notation d and d of 4.4. Next, the following result holds. Lemma 5. We have j j!j! s j Ds, s 2 s d log j 2 N j+j D0, 0. 2 s=s 2=0 Proof. Let d =a d. Analogously to , we have, for s, s 2 η 0, log D 0 s, s 2 = d W s W s +s 2 a W W s W = d W s +s 2 W W s a d W W s Kη 0 +d 7.5 K d +d K +d and s log G s, s 2 = p V p V K p V d, log p p +s log p p δ d p s2 p d +s p + +s K p s 2 +d p δ2 η 0 log 2 p log p p δ +d p δ p V V δ Kη 0 log V +d log V Kη 0 log 2 2 N +d log 2 N d log 2 N. The treatment of G 2 will be similar to this and 7.0. By d a p +s ν p ν 3 p = ν p H ν p H p H 2 p H \H 2 = a d, 7.6

21 primes in tuples ii 2 we have log G 2 s, s 2 s = p p>v p p>v d d log p p +s ν p ν 3 p+ν 3 p p s2 ν pp +s ν 2 pp +s2 +ν 3 pp +s+s2 d d p +s p +s p s2 p +s log p p δ d +Kp η0 p log p p δ +Kη 0 p +o log p p e 2/η 0 log p p δ +Kη 0 log 2 p p δ +K log p e /η0 p p +o log d log 2 +Kη 0 log log N K 3 d +Kη 0 log 2 N log 2 N d log 2 N. log 2 p K log + p δ log N d Finally we have, similarly to above and 7., using a dp s2 =a d+d p s2, log G 3 s, s 2 s log p a p +s = p p>v p>v p +s+s2 p +s +a dp s2 ap +s bp +s2 +dp +s+s2 log p K p δ p δ + K p δ d +Kp η0 Kd p>v log p p 2 2δ +K2 η 0 V <p e /η 0 K 2 log 2 p +K2 p2 2δ Kd V 2δ + K2 η 0 log V V 2δ + log N 2δ η0 d K V d +Kη 0 log 2 N+o d +Kη 0 log 2 N d. p>e /η 0 log p p 2 2δ η0 p +s+s

22 22 d. a. goldston, j. pintz and c. y. yildirim Now, imply, by symmetry for j =, 2, that log Ds, s 2 s j for s, s 2 η 0, 7.9 η 0 and therefore, similarly to 7.2, we have which by 7.2 proves Lemma 5. max Ds s η0 s 2 η0, s 2 D0, 0, Evaluation of the integral I This section will be devoted to the examination of I,, the sum of the residues in The rather complicated formula 6.29 yields the main term and all secondary terms of the form log R m exclusively for m [d, d+u+v ] and will additionally contribute to other secondary terms for m [0, d ]. However, from the terms I, j, j, ν belonging to the triplet j, j, ν in the triple summation, only those with ν=0 and j =0 contribute to the main term of order log R d+u+v, since in all other terms the exponent of log R is d+u+v j ν. We now have to work more carefully than in [7]. For example, by the aid of Lemma 2 a generalization of 8.6 in [7] we will exactly evaluate the coefficients A j,ν of j!ν! ν j s ν 2 in 6.29 as follows. Let j, ν 0, s j where we may assume, by 6.29, that Ds, s 2 log R v+d+u j ν m := j j 0 and y := v+d ν, 8. ν v+d+m, that is m ν v d = y. 8.2 Then we have from 6.29, by notation 5.6, 8. and Lemma 2, A j,ν = j u j!ν! u m+j m dd+... d+m u! m+j j v+d+m ν!ν! = u j m=0 m y m=0 m y m dd+... d+m u j m!m! v+d+m ν! = Zd, u j, v+d ν 8.3 = v ν+... v ν+u j u j!d+v ν+u j!. We have to compare A j,ν with A 0,0. This will be furnished by the following lemma.

23 primes in tuples ii 23 where Lemma 6. We have Proof. Recalling 6.24, we have A j,ν := A j,ν A 0,0 +ν CKj. A d+v+u! u j +... u j,ν = d+v+u ν j! v+u j +... v+u A j,ν CK j +ν A j,ν, A j,ν = v ν+... v ν+u j v+... v+u j. 8.4 If ν 2v+, then clearly A j,ν, so we may suppose that ν = Bv+, B > In this case we have, by u j u v<v+, ν u j A j,ν B v+ = B ν/b < 2 ν, 8.6 v+ since the maximum of x /x in [, is attained at x=e and e /e <2. Now we are ready to evaluate the crucial term I, by the aid of Lemmas 2, 5 and 6. Namely, by 4. and 4.4, R N c, 6.29, 8.3 and 6.24, we have I, = A 0,0 log R d+u+v D0, 0+ u j =0 v+d+u j ν=0 j +ν = Zd, u, v+dlog R d+u+v D0, 0 +O = d+v+u! v+u u log R d+v+u D0, 0 A j,ν log R j +ν j =0 ν=0 j +ν j!ν! K +O d log 2 N log R j s j ν s ν 2 D0, 0 K d j log 2 N +ν log R. 8.7 The integral I,2 in 6.29 does not contribute to the main term and can be estimated relatively easily due to the presence of the term R s2 s 2 L 4. In fact, choosing η = 00log 2 N +log T as earlier, we obtain by Lemma 4, 5.2, 6.20 and for any s 2 L 4, Zs 2 u j j=0 j =0 log R u j CK j j u j!j j! e C K log 2 N+K log 3 N log t K+O CKlog 2 N +log T D0, s 2 R σ2 j s 2 d+j j +v+ s 2 b+o K K R σ2. 8.8

24 24 d. a. goldston, j. pintz and c. y. yildirim Now Lemma yields immediately, by 6.24 and R N c, I,2 = Zs 2 ds 2 e C K log2 N+K log 3 N c log N e c log N πi L 4 We may summarize 8.7 and 8.9, by D0, 0=D 0 0, 0G0, 0=G0, 0 0 which is true by the admissibility condition, by and 6.23, as the following. Lemma 7. The integral I in 6.26 satisfies the asymptotic v+u K I = log R d+v+u G0, 0 +O d log 2 N d+v+u! u log R +O e c log N Estimate of the integral I 2 For I 2 in 6.27, after interchange of the two integrations, we move the contour L 2 for the inner integral over s 2 to the left to L 4 passing a pole of order d at s 2 = s if t 2 U and a pole of order v+ at s 2 =0, and obtain I 2 = Ds, s 2 R s+s2 Res 2πi s 2= s L 3 + 2πi + 2πi 2 s u+ s v+ 2 s +s 2 d Ds, s 2 R s+s2 Res L s 3 2=0 s u+ s v+ 2 s +s 2 d R s F s, s 2 L 3 s a+u+ L 4 =: I 2, +I 2,2 +I 2,3 +O e c log N. ds ds R s2 s b+v+ ds ds 2 +O e c log N 2 9. By the argument of Lemma and 6.2, the third integral I 2,3 is e c log N. The second integral I 2,2 is completely analogous to I,2 in 6.29, which was estimated by e c log N in 8.9, the only change being that the roles of s and s 2 are interchanged. where The residue in I 2, is zero if d=0, while for d we have Ds, s 2 R s+s2 d Ds, s 2 R s+s2 = lim s 2! s Res s 2= s s u+ s v+ 2 s +s 2 d d B j s, H, H 2 = j j d! s d 2 s u+ s v+ 2 = d B j s, H, H 2 log R d j, d! j=0 9.2 j j ν Ds ν s j ν, s 2 ν v+... v+ν 2 s2= s ν+v+ s u+v+ν ν=0

25 primes in tuples ii 25 We thus obtain where I 2, = d C j H, H 2 log R d j, 9.4 d! j =0 C j H, H 2 = B j s, H, H 2 ds, j = 0,, 2,..., d πi L 3 It remains to estimate these quantities, which are independent of R. We are allowed to transform the contour L 3 in 9.5 to the contour L, defined in 5.4. Our task is now the estimation of the integral C j on the new contour L =L 0 L, since the integral on the horizontal segments t =U is O e c log N. 0. Comparison of Ds, s and D0, 0 We have seen in 7 that by Lemma 4 we can estimate j s j j s j 2 Ds, s 2 with the aid of Ds, s 2. We will show now how to estimate Ds, s /D0, 0 from above when s is on the contour L, which, together with Lemma 0, will play a crucial role in the estimation of I 2,, which is the main part of I 2. First we note that if s L is on the semicircle L 0, then by 7.2 we obtain Ds, s e C K D0, 0, s L Thus, in the following, we may suppose that s = it, t > 0, 0.2 since D it, it = Dit, it. Our main results in this section are the following estimates. Lemma 8. We have, for any real t, Dit, it max{, t } a+b/2 D0, 0. Together with 0., this implies the following result. Lemma 9. We have, for s L, where L was defined in 5.4. Ds, s e C K max{, t } a+b/2 D0, 0,

26 26 d. a. goldston, j. pintz and c. y. yildirim From 6.9 and 6.23 we have D=D 0 G G 4. We first examine the behavior of the functions D 0 s, s and G s, s on the imaginary axis, which requires a lemma concerning W s from Lemma 0. There exist positive absolute constants t 0 and t, with t >, such that W it e t2 /6 = W 0 for t t 0, 0.3 W it t 2/3 for t t. 0.4 Proof. We will use that in a neighborhood of s=0 we have, for the entire function W s, the representation W s = +γ 0 s+ γ ν s ν+, 0.5 where γ 0 =γ is Euler s constant and see [2, notes on p. 49] γ 0 = γ = , γ = This implies that ν= W it 2 = W itw it = +iγt γ t 2 +Ot 3 iγt γ t 2 +Ot 3 = +t 2 γ 2 2γ +Ot 3, 0.7 as t!0. Now prove 0.3 for t t 0, while 0.4 clearly holds by 5.2. Remark. If W it for any t which could be checked by computers, since t 0 and t are explicitly calculable, then the following simple lemma is not necessary. Lemma. Given any positive constants B 0, B and ε, we have, for any t [B 0, B ] and any X >CB 0, B, ε, Jt, X := p X p it p cb 0, B log X /2 ε. 0.8 Proof. Fix t. Since every factor is at least, we can neglect those with cost log p>0. On the other hand, if cost log p 0, then we have p it log p > log p > p. 0.9 The primes satisfying cost log p 0 are in intervals of the type I j = [e π2j+/2/t, e π2j+3/2/t ] =: [e mj, e mj+π/t ] 0.0

27 primes in tuples ii 27 and I j [, X] if 2π j+ 3 4 /t log X, that is, if j t log X 2π 3 4 =: j. 0. Using the prime number theorem, we obtain by partial summation Hence, by 0.9, we have which completes the proof. e m j +π/t p dx p I e m j x log x = log 2j j+ = 2j +O j j log Jt, X > ε ε +O > 2j 2 log 2 X c B 0, B, 0.3 j j Remark. Working more carefully, we could prove Lemma with log X /2 ε replaced by log X, but the result we just obtained suffices for our needs. Taking into account the trivial relation we obtain, from Lemmas 0 and, the following result. p it p, 0.4 Lemma 2. We have, with a sufficiently small constant t 0 <, E 0 t := D 0 it, itg it, it D 0 0, 0G 0, 0 /6, if t t e a+bt2 0, 0.5 and for any t >t 0 and N >N 0, E 0 t e a+b max{, t } a+b/ Proof. By 0.4 and the definition of D 0 in 6.23, we clearly have G it, it G 0, 0, 0.7 and D 0 it, it = W it a+b, D 0 0, 0 = W 0 =, 0.8 which immediately imply E 0 t W it a+b. 0.9

28 28 d. a. goldston, j. pintz and c. y. yildirim Hence, by Lemma 0, we have 0.5, and 0.6 for t t, where t is a sufficiently large constant. Actually t >e 6 is sufficient. Letting we have, by Lemma, Ct 0, t := max t 0 t t W it, 0.20 E 0 t = J t, V W it a+b ct0, t log V /3 Ct 0, t c log 2 N a+b/3 et a+b, a+b 0.2 which proves 0.6. We will continue our study of Dit, it with that of L 4 t := log G 4it, it G 4 0, 0 = Re log G 4it, it G 4 0, We first divide each term by +ν 3 p/p /p d in the product representation 6. of both G 4 it, it and G 4 0, 0. After this, we take the logarithm of each term and use the formula log z = z+ z m m m=2, if z < We now separate the effect of the linear terms and those of order m 2 and write accordingly L 4 t = L 4, t+l 4,2 t We have, by the trivial relations ν p a and ν 2 p b, L 4, t = a+b p ν p+ν 2 p cost log p p+ν 3 p/p p>v We remark that the sum is convergent, since ν p=a and ν 2 p=b for p. The contribution of the higher-order terms of G 4 to L 4,2 which do not involve the functions ν j p are majorized for any t by a+b p>v m=2 while for small t they are majorized by 2 Ca+b mpm V log V C log 3 2 N, 0.26 Ca+bt 2 p>v log 2 p p 2 Ct2 log 2 N. 0.27

29 primes in tuples ii 29 Similarly, using 3., 4. and 6.24, the contribution of the higher-order terms of G 4 to L 4,2 which involve the functions ν j p are majorized for any t by p>v m=2 2a+b m mp m Ca+b2 V log V Ca+b log 3 2 N, 0.28 while for small values of t they are majorized by m m ν p j ν 2 p m j j mp+ν p>v m=2 3 p m cosm 2j t log p j =0 C a+b m mp m t2 m 2 log 2 p p>v m=2 Ca+b 2 t 2 p>v Ca+b2 t 2 log V V Ca+bt2 log 2 N. log 2 p p 2 Summarizing , we have proved the following result. Lemma 3. We have G 4 it, it G 4 0, 0 e Ca+b min{,t2 }/log 2 N Comparing the above with , we see that remain valid if we multiply them by G 4 it, it/g0, 0. This proves Lemma 8.. Estimate of I 2, and evaluation of I In this section we will estimate the integral I 2, based on formulas , using Lemmas 4 and 9. We first obtain from the above lemmas for s L, as j d min{a, b} K, v K and a+b K, j B j s, H, H 2 dj s u+v+2 Ds, s CKlog 2 N +log T j ν ν=0 K ec D0, 0d j log t +3 j max {, t } a+b δ u+v 0 s 2 j ν j= +v/j s CK log 2 N j ν K log 2 N ν e C K CK 2 log 2 N j δ u+v 0 s 2 D0, 0.. ν=0

30 30 d. a. goldston, j. pintz and c. y. yildirim Integrating the above upper bound along L, we obtain C j H, H 2 e C K CK 2 log 2 N j δ u+v+ 0 D0, 0..2 Finally, summation over j d yields in 9.4, as R N c and u+v K, K I 2, ec D0, 0δ u+v+ 0 log R d d! d CK 2 log 2 N log R j =0 ecu+v D0, 0log R d K log 2 N u+v+ d! D0, 0log Rd+u+v CK 3/2 u+v+ log 2 N d+u+v! log R D0, 0log Rd+u+v log N K/50. d+u+v! j.3 This implies, by 9. and 9.4, that I 2 D0, 0log Rd+u+v log N K/50 +e c log N..4 d+u+v! This yields, by Lemma 7, the final asymptotic evaluation of I in 6.22, since D0, 0=G0, 0, as v+u K I = log R d+v+u G0, 0 +O d log 2 N d+v+u! u log R where G0, 0 = p P thereby proving Theorem 4. ν ph p p +O e c log N,.5 H = SHP p AH,.6 2. A Bombieri Vinogradov type theorem In the present section we will prove a modified Bombieri Vinogradov theorem, where the examined moduli are all multiples of a single modulus M. It would facilitate our task if we were entitled to use the following hypothesis. Hypothesis SY. If L δ, χ=0 for δ>0 and a real primitive character χ mod q, q Y, then δ 3 log Y for an explicitly calculable absolute constant Y >C 0. 2.

31 primes in tuples ii 3 We note that we have the effective unconditional estimate [6], [5], valid for q>q 0 : δ q. 2.2 A further observation similar to that of Maier [3] is that by the Landau Page theorem cf. Davenport [2, 4], with some constant c in place of 3, or Pintz [6] with 2 +o for any given Y there is at most one real primitive character χ which does not fulfill 2.. This makes it possible to turn hypothesis SY into a theorem, valid for a sequence Y =Y n! for n>n 0, an explicitly calculable absolute constant with Y n e Yn. 2.3 In order to show this, suppose that 2. is false for a sufficiently large Y, i.e. by 2.2 there exists a character χ mod q Y such that L δ, χ =0 with Let us choose Ỹ >Y in such a way that { } min, c 0 δ < Y q 3 log Y. 2.4 Ỹ = e /3δ, that is δ = 3 log Ỹ. 2.5 Then, for any other zero δ 2 belonging to a real primitive character χ 2 mod q 2, q 2 Ỹ, we have by the Landau Page theorem in the version of Pintz [6], max{δ, δ 2 } > 3 log Ỹ, that is δ 2 > 3 log Ỹ. 2.6 Now, show that 2. is true for a value Y =Ỹ satisfying We can formulate this in the following way. Y < Ỹ < e Y / Lemma 4. Hypothesis SY holds for a sequence Y n! with Y n e Yn, 2.8 where Y 0 can be chosen with Y 0 <C 0, an explicitly calculable absolute constant. An alternative to this lemma and this approach would be to use Heath-Brown s theorem [0] but only in case of Theorem according to which either i SY holds for every Y >C, with some absolute constant C, or ii there are infinitely many twin primes.

32 32 d. a. goldston, j. pintz and c. y. yildirim The significance of the real zeros in hypothesis SY is that a similar inequality holds with Re ϱ in place of δ and with a constant c 0 in place of 3 if Im ϱ is not too large; this is the standard zero-free region of L-functions cf. Davenport [2, 4]. Lemma 5. There exists an explicitly calculable absolute constant c 0 < 3 Ls, χ 0 in the region apart from possible real exceptional zeros of real L-functions. c 0 such that σ > log q t +3, 2.9 Now we are in a position to formulate and prove the following theorem which is similar to but stronger than Lemma 6 of Maier [3]. Theorem 6. Let c be an arbitrary, fixed positive constant. strictly monotonically increasing continuous function of X with Let Y =Y X be a e 2 log X Y X X. 2.0 Then there exists a sequence X n! satisfying X <C 0, an explicitly calculable constant, with the following property. Let X =X n, L=log X, M min { 4 Y X, X /8 } be a natural number, and E X, q := max x X max a,q= Q = X /2 M 3 e c log X, 2. EX, q, a := max x X max a,q= p x p a mod q log p x ϕq. 2.2 Then q Q q,m= E X, Mq X M L5 e c2 log X/log Y X, 2.3 where c 2 =min { 6 c, 4 c 0}. i.e. Remark. The above theorem holds with any X, for which SY X=SY is true, Ls, χ 0 for s ] 3 log Y, holds without exception for all real primitive characters χ mod q, where 2.4 q Y = Y X. 2.5

33 primes in tuples ii 33 Proof. We will choose our sequence X n =Y Y n, where Y n is the sequence supplied by Lemma 4 for which SY is true and Y is the inverse function of Y X. Alternatively, if hold, then we can choose X as an arbitrary sufficiently large number. In both cases SY, i.e , hold. Using the explicit formula for primes in arithmetic progressions with T = X log 2 X ϱ=β+iγ= δ+iγ denotes a generic zero of an L-function we obtain cf. Davenport [2, 9] for any a with a, q=, q Q and y X the relation Ey, q, a = ϕq χa χq ϱ=ϱ χ β /2 γ T y ϱ ϱ +OL2 y. 2.6 The effect of the last error term is clearly suitable, O Q L 2 X in total. We can classify zeros of all primitive L-functions mod q M Q M M M up to height T into OL 4 classes B, λ, µ, ν by Lemma 5, as [ [ Mλ M 2, M Qν λ, q 2, Q ν where, γ M λ = 2 λ 2M, Q ν = 2 ν 2Q, T µ = 2 µ 2T and [ [ Tµ 2, T c0 µ, δ L, +c 0, 2.7 L c 0 L 2, 2.8 with the additional class of index 0: γ [0, =[0, T 0. The set of quadruples, λ, µ, ν satisfying 2.8 with ν, µ 0, λ and 0 will be denoted by B. where In this case we have clearly by 2.6, similarly to Davenport [2, 28], q Q q,m= E X, qm X M L6 max,λ,µ,ν B N σ, Q, T = N +c 0 /L, M λ Q ν, T µ Q ν T µ X c0 /L, Q/2<q Q χq ϱ=ϱ χ χ primitive β σ γ T We will see that, in order to prove our theorem, it will be enough to prove, for any quadruple δ, M, Q, T with the property cf. 2.9 and that c 0 log QM T δ 2, 2 M M, 2 Q Q, T T, or c 0 log Y δ 2, 2 M M, Q Y, T = T 0 =, δ 2, 2 M M, Q > Y, T = T 0 =,

34 34 d. a. goldston, j. pintz and c. y. yildirim the crucial inequality N δ, M Q, T L 9 QT X δ e c log X/log Y 2.22 with some positive absolute constant c. The first line in 2.2 is meant to cover all non-real zeros, the second and third lines are meant to cover the real zeros. We will use Theorem 2.2 of Montgomery [4]: N δ, Q, T Q 2 T 3δ/+δ log QT We do not need, for the range δ 5, the stronger inequality of Theorem 2.2 of [4] with the exponent 3δ/+δ replaced by the smaller 2δ/ δ. Since 3δ/+δ, 2.22 will follow if we can show that M 6δ Q 6δ/+δ X δ e c2 log X with c 2 = 6 c As in the range 0 δ 2 we have 6δ/+δ 2δ, this is true by the definition Q = X /2 M 3 e c log X, if δ 2. In case δ 2 we have, by 2.23, N δ, M Q, T QT /2 M 6δ log If we have here QT e X, then 2.25 directly implies 2.22, since N δ, M Q, T QT QT /2 M 6δ X δ e log X/ If δ 2 and QT e log X Y, then we may assume δ c 0 /log MQT by 2.9 or δ /3 log Y >c 0 /log Y by , since the modulus of the corresponding primitive character is q M 4QM Y. Hence, M 6 X δ X δ/4 exp c { 0 log X 8 min log QT, log X } log M, log X log = e c0 log X/8 log Y. Y Remark. The condition for Q could be weakened to Q <X /2 M e c log X, but this has no significance in our application.

35 primes in tuples ii Proof of Theorem 5 The method of proof of Theorem 5 is quite similar to that of Theorem 4. The basic difference is that instead of the trivial problem of the distribution of integers in arithmetic progressions we have to use properties of the distribution of primes in arithmetic progressions. Since we have to consider the weighted sum of the error terms in the formula for the number of primes in arithmetic progressions, the Bombieri Vinogradov theorem can help us. However, due to the relatively weak estimate of the original Bombieri Vinogradov theorem, it does not lead to better results than p n+ p n lim inf = 0. n! log p n That is partly why we need to use Theorem 6 instead. Our situation is even more complicated here, since we need the moduli of the progressions to be multiples of a number V. Fortunately our present Theorem 6 solves this problem in a completely satisfactory way, even without loss if P =M e +o log N which is now the case by V = log N. We will suppose that N = 3 X n, n is sufficiently large, M =P and Y X=e 3 log X in Theorem 6. If we use Heath-Brown s theorem [0] we may assume hypothesis SY for any N, and then N can be an arbitrary, sufficiently large integer. In the course of the proof we will follow closely the analogous proofs of Propositions 4 and 5 in 7 9 of [7], so we will sometimes omit details. Let x Θx; q, a := log p = [a, q = ] +Ex; q, a, 3. φq p x p a mod q where [S] is if the statement S is true and 0 if S is false. We have for a regular residue class ã with respect to H and P, S R N; H, H 2, l, l 2, P, ã, h 0 := = 2N n=n+ Λ R n; H, l Λ R n; H 2, l 2 θn+h 0 K +l!k +l 2! log R d d,e R K+l log R e µdµe K+l2 n N n ã mod P d P H n e P H2 n θn+h For the inner sum, we let d=a a 2 and e=a 2 a 2, where d, e=a 2, and thus a, a 2 and a 2 are pairwise relatively prime. In the following, we may suppose that d, P = e, P =,

36 36 d. a. goldston, j. pintz and c. y. yildirim otherwise the last sum would be zero, since by the regularity of ã we have P H n, P = P H2 n, P =. The n for which d P H n and e P H2 n cover certain residue classes modulo [d, e]. If n b mod a a 2 a 2 is such a residue class, then letting m=n+h 0 b +h 0 mod a a 2 a 2, b b mod a a 2 a 2 and b ã mod P, we see that this residue class contributes to the inner sum θm N++h 0 m 2N+h 0 m b+h 0 mod a a 2 a 2 P = θ2n +h 0 ; a a 2 a 2 P, b+h 0 θn +h 0 ; a a 2 a 2 P, b+h 0 N = [b+h 0, a a 2 a 2 P = ] φa a 2 a 2 P +OE 3N, a a 2 a 2 P. 3.3 We need to determine the number of these residue classes where b+h 0, a a 2 a 2 P = so that the main term is non-zero. The condition ã+h 0, P =b+h 0, P = is equivalent to ã being regular with respect to H 0, since ã is regular with respect to H. Thus we will assume from now on that ã is regular with respect to H 0. If p a then b h j mod p for some h j H, and therefore b+h 0 h 0 h j mod p. Thus, if h 0 is distinct modulo p from all the h j H, then all ν p H residue classes satisfy the relatively prime condition, while otherwise h 0 h j mod p for some h j H leaving ν p H residue classes with a non-zero main term. We introduce the notation νph for this number in either case, where we define for a set G, νpg = ν p G 0, 3.4 with G 0 = G {h 0 }. 3.5 We extend this definition to ν d H for squarefree numbers d, by multiplicativity. The function ν d is familiar in sieve theory, see [8]. The same applies for ν d H 2 and ν d H H 2, as in 6.2. Since En; q, a log N if a, q> and q N, we conclude that N+ n 2N n ã P d P H n e P H2 n θn+h 0 = νa H νa 2 H 2 ν a N 2 H H 2 φa a 2 a 2 P +Od K a a 2 a 2 E 3N; a a 2 a 2 P. 3.6