ψ(y + h; q, a) ψ(y; q, a) Λ(n), (1.2) denotes a sum over a set of reduced residues modulo q. We shall assume throughout x 2, 1 q x, 1 h x, (1.

Transcription

1 PRIMES IN SHORT SEGMENTS OF ARITHMETIC PROGRESSIONS D. A. Goldston and C. Y. Yıldırım. Introduction In this paper we study the mean suare distribution of primes in short segments of arithmetic progressions. Specifically we eamine I(, h, ) = * a() ( ψ(y + h;, a) ψ(y;, a) h ) dy (.) φ() where ψ(;, a) = n n a() Λ(n), (.) Λ is the von Mangoldt function, and * a() denotes a sum over a set of reduced residues modulo. We shall assume throughout,, h, (.3) the other ranges being without interest. As far as we are aware the only known result concerning the general function I(, h, ) is due to Prachar [], who showed that, assuming the Generalized Riemann Hypothesis (GRH) I(, h, ) hlog. (.4) On the other hand, much more is known about the special cases where one of the two aspects, segment or progression, is trivialized. Indeed, our function I(, h, ) is essentially a hybrid of the more familiar functions I(, h) = the second moment for primes in short intervals, and G(, ) = * a() (ψ(y + h) ψ(y) h) dy, (.5) ( ψ(;, a) ), (.6) φ() 99 Mathematics Subject Classification. Primary:M6. Supported by an NSF Grant The second author was on a sabbatical at San Jose State University Typeset by AMS-TEX

2 GOLDSTON AND YILDIRIM which measures the total variance in the prime number theorem for arithmetic progressions modulo. We see I(, h, ) = I(, h) so that I(, h, ) generalizes I(, h). We shall see for small that I(, h, ) behaves rather similarly to I(, h). When h is about then h I(,, h) will behave rather similarly to G(, ). In studying I(, h, ) we will use some techniues from the recent papers Goldston [4] and Friedlander and Goldston [] where I(, h) and G(, ) were eamined. It is helpful in understanding the results we obtain to first consider what follows from the Riemann Hypothesis (RH) and a strong form of the twin prime conjecture. Let N = N (k) = ma(0, k), N = N (, k) = min(, k), (.7) and E(, k) = where we define as usual S(k) = N (k)<n N (,k) C p k p> Λ(n)Λ(n + k) S(k)( k ), (.8) ( ) p, if k is even, k 0; p 0, if k is odd; (.9) with C = p> ( ) (p ). (.0) Theorem. Assume the Riemann Hypothesis and that for 0 < k and some given ǫ (0, 4 ) E(, k) +ǫ. (.) Then for h/ ǫ and h we have I(, h, ) hlog ( ). (.) h In the smaller range 4ǫ h/ ǫ we have I(, h, ) = hlog( h ) h γ + log π + log p +O(h )+O ǫ (h ǫ ). (.3) p In the case = we recover the formula I(, h) hlog(/h), h ǫ (.4) subject to the same hypotheses. We may conjecture that euation (.4) hold for the epanded range h ǫ, since Goldston and Montgomery [5] showed this conjecture is euivalent under the RH to a pair correlation conjecture for zeros of the Riemann zeta-function. Therefore we might conjecture the condition h/ ǫ for (.) might be relaed. In the case of G(, ) it was proved in [] that under the same conjectures G(, ) log, +ǫ. (.5)

3 PRIMES IN ARITHMETIC PROGRESSIONS 3 We mention that the case h/ may be dealt with trivially, and it is easy to show that I(, h, ) hlog, h. (.6) The conjecture (.) is a very strong conjecture and one purpose of this paper is to see what can be proved when we replace this conjecture with GRH. For certain small ranges of h and our results are unconditionally true, but the results conditional on GRH are more interesting. We begin by proving that (.) and (.3) holds for almost all in a smaller range. Theorem. Assume the Generalized Riemann Hypothesis. Then we have for h 3/4 Q h that I(, h, ) hlog( h ) + h γ + log π + log p p Q/< Q h 7 4 log min(h Q 3 log 3 Q, hq) + h (Q + log 3 ). (.7) From this theorem we obtain the following almost-all result, where we mean by almost-all that all ecept at most o(q) integers in the interval [Q/, Q] satisfy the given property. Corollary. Assume the Generalized Riemann Hypothesis. Then for almost all with h 3/4 log 5 h we have I(, h, ) hlog( h ) (.8) and for h = o() in the range h 3/4 log 5 o(h/ log 3 h) we have I(, h, ) hlog( h ) h γ + log π + log p. (.9) p The range where the Corollary holds is h h 4 log 5, (.0) which is smaller than the range of validity in Theorem. Net we prove a Barban-Davenport-Halberstam type theorem for I(, h, ). Theorem 3. Assume the Generalized Riemann Hypothesis. Then we have, for Q h, Q I(, h, ) = Qhlog( Q h ) cqh + O(min(Q 3 h log 3 Q, Qh) + O(Qh ) + O(h 3 log 6 ), (.)

4 4 GOLDSTON AND YILDIRIM where c = γ + log π + p log p p(p ). We thus obtain an asymptotic formula I(, h, ) Qhlog( Q h ) (.) Q provided h log 6 Q h. This gives a range of validity h h log 6 (.3) which is larger than the range in Theorem and close to Theorem when h is close to. All of our results in Theorems, and 3 contain an error term containing an O(h ) which is significant as a second order term if h is close to or eual to. This error term can be replaced by an eplicit epression; we have chosen not to do so to keep the appearance of the results as simple as possible. We have retained these terms in Lemma 4, and it is straightforward to retain them through the paper and obtain the complete second order terms for h close to. Finally we prove that we can obtain reasonable lower bounds for I(, h, ) consistent with (.7). Theorem 4. Assume the Generalized Riemann Hypothesis. Then for any ǫ > 0 and h 3, we have ǫ log 3 I(, h, ) h ( ( log ) ) 3 O(h(log log ) 3 ) (.4) h Letting h = α, we have in particular for any ǫ > 0 and 0 α 3, ( I(, h, ) 3 ) α ǫ h log. (.5) Euation (.5) improves the result obtained in [4]. This improvement is based on a suggestion of Heath-Brown. A similar improvement has been made in [3] for the result in []. Our results have interesting connections with the pair correlation of zeros of Dirichlet L-functions, and allow us to obtain new result that connect and etend the earlier work of Yıldırım [] and relate it to work of Özlük [0]. We will present these results in a later paper.. Preliminaries and Lemmas In this section we relate I(, h, ) to E(, k) and obtain the necessary lemmas for proving our results. We have I(, h, ) = * a() h φ() (ψ(y + h;, a) ψ(y;, a)) dy * a() (ψ(y + h;, a) ψ(y;, a)) dy + h φ() =S h φ() S + h φ(). (.)

5 PRIMES IN ARITHMETIC PROGRESSIONS 5 Now S = * a() n a() = Λ(n)f(n,, h), n (n,)= Λ(n) dy [,] [n h,n) where n, for n < + h h, for + h n f(n,, h) = dy = [,] [n h,n) n + h, for < n + h 0, elsewhere. (.) Since we conclude n +h (n,)> S = Λ(n) = <n +h ν p ν +h log p log p log, (.3) Λ(n)f(n,, h) + O(h log ). (.4) To evaluate sums involving f(n,, h) we use the following result. Lemma. Let C() = n c n. Then we have c n f(n,, h) = +h C(u)du <n +h where c v = 0 if v is not an integer. Proof. Since the left-hand side is +h f(u,, h)dc(u) = = the lemma follows. +h +h +h C(u)d u f(u,, h) C(u)du + +h C(u) du (.5) C(u)du, Now writing R() = ψ(), (.6) we obtain on taking C() = ψ() in Lemma that S = h + +h R(u)du To evaluate S, we use the following lemma. +h R(u) du. (.7)

6 6 GOLDSTON AND YILDIRIM Lemma. For real numbers a n and b n we have a n b m dy = a n b n f(n,, h) y<n y+h y<m y+h <n +h + (a n b n+k + a n+k b n )f(n,, h k). 0<k h <n +h k (.8) Proof. The left-hand side of (.8) is = <m,n +h m n h a n b m [,] [n h,n) [m h,m) dy. The terms n = m give the first term on the right of (.8). The terms with n < m are = a n b m f(n,, h (m n)), <n<m +h m n h and letting m = n + k this becomes 0<k h <n +h k a n b n+k f(n,, h k). The terms m < n contribute the symmetric term in (.8). S = By Lemma we see that <n +h (n,)= Λ (n)f(n,, h) + 0<k h k 0() <n n+h k (n(n+k),)= Λ(n)Λ(n + k)f(n,, h k). A calculation similar to (.3) shows that we may drop the conditions (n, ) = and (n(n + k), ) = in the above sums with an error h log + h log. (.9) Thus we have S = S 3 + S 4 + O(( + h )h log ), (.0) where and S 3 = Λ (n)f(n,, h) (.) <n +h S 4 = Λ(n)Λ(n + j)f(n,, h j). (.) 0<j h/ <n +h j

7 To evaluate S 3 we let PRIMES IN ARITHMETIC PROGRESSIONS 7 P() = n Λ (n) log + (.3) and on applying Lemma with C() = n Λ (n) we obtain ( + h) S 3 = + log( + h) () ( + h) log log( + h) log 3h + +h P(u)du +h P(u)du =hlog + log( ( + h ) ) + h( 3 ( + h ) + log 4( + h ) ( + h ) ) + h log( + h + h ) + +h P(u)du +h P(u)du. (.4) For S 4, we take C(u) = 0<n u Λ(n)Λ(n + j) in Lemma and obtain S 4 = = {( +h +h +h j 0<j u 0<j u +h j n u j n u j ) n uλ(n)λ(n + j)du} Λ(n)Λ(n + j)du On combining our results on S, S, and S 3 we obtain Λ(n)Λ(n + j) du. (.5) I(, h, ) =hlog + log( ( + h ) ) + h( 3 ( + h ) + log 4( + h ) ( + h ) ) + h log( + h + h ) + +h 0<j u +h n u j ( h +h R(u)du φ() + +h P(u)du +h 0<j u n u j Λ(n)Λ(n + j)du h φ() +h R(u) du ) Λ(n)Λ(n + j)du P(u)du + O( h log ) + O(h log ). (.6) We see that if h/ the double sums above vanish and by the prime number theorem the terms with R(u) and P(u) contribute h( + h φ() ) log A. Thus

8 8 GOLDSTON AND YILDIRIM euation (.6) follows from (.6). To evaluate the double sums over primes in (.6), we use (.8) and find +h +h = 0<j u 0<j h/ +h 0<j u n u j n u j (h j)s(j) + 0<j u Λ(n)Λ(n + j)du Λ(n)Λ(n + j)du +h 0<j u E(u, j)du E(u, j) du. (.7) To evaluate the singular series above we using the following result from []. Lemma 3. We have (y j)s(j) = y j y ( φ() y log y y γ+log π + for any δ, 0 < δ <, where, letting = (,), I δ (y, ) = S( ) πi δ+i δ i ζ(s) p ) log p +I δ (y, ), (.8) p ( )( y ) s+ ds + (p )p s s(s + ). Further, we have the estimates I δ (y, ) S()y δ δ ( ( δ) 3 + )( p δ and, for arbitrarily small fied ǫ, η > 0, and min 0<δ< + ), (.9) p ( δ) ( ) min I δ (y, ) min(y(log log 3) 3, y log y ep η ), (.0) 0<δ< log Q/< Q I δ ( h, ) min(q 3 h log 3 Q, Qh), (h ǫ Q h). (.) On combining (.6) with (.7) and (.8) we obtain the following result which we use in the proof of Theorems,, and 3.

9 PRIMES IN ARITHMETIC PROGRESSIONS 9 Lemma 4. We have I(, h, ) =hlog( h ) h γ + log π + + log p p log ( + h ) ( + h ) + log( ( + h ) ) + h ( + h ) log( + h + h ) + I δ( h, )) +h +h + E(u, j)du E(u, j)du 0<j u ( h +h R(u)du φ() + +h P(u)du +h +h R(u) du ) 0<j u P(u)du + O( h log ) + O(h log ). (.) 3. Proof of Theorems and. Assuming the Riemann Hypothesis, we have [] and by partial summation R() log, (3.) P() log 3. (3.) Using Lemma 4 and these estimates we find after epanding the logarithmic terms into power series that, assuming RH, I(, h, ) =hlog( h ) h γ + log π + log p p Since + O(h ) + I δ ( h, ) +h + 0<j u +h E(u, j)du 0<j u E(u, j)du + O( h / log ) + O(h / log 3 ). (3.3) φ() log p p p log log p p log log 3, (3.4) on using the former bound in (.0), and the conjectured bound (.) we see that (.) holds. In order to prove (.3) we need to show that min I δ ( h 0<δ<, ) h ǫ. (3.5)

10 0 GOLDSTON AND YILDIRIM If > e η (η will be specified in terms of ǫ below), then the latter bound in (.0) yields min I δ ( h 0<δ<, ) (h) +η. (3.6) For 4ǫ h ǫ and ǫ < 4 we have (h) +η (h (+ +4ǫ ) +η ǫ ( = h +4ǫ )(+η) h ǫ if we choose η = ǫ 3. For e η, we appeal to (.9) with δ = ǫ to see that (3.5) holds with the implied constant now depending on ǫ. We now turn to the proof of Theorem, which depends on the following result essentially due to Kaczorowski, Perelli, and Pintz [7]. Proposition. Assume the Generalized Riemann Hypothesis. Then, with H N, we have uniformly in N that N k N+H E(, k) H log. (3.7) We first derive Theorem, before making some comments on the proof of Proposition. By (3.3) we have Q/< Q I(, h, ) hlog( h ) + h γ + log π + log p p h Q + I δ ( h, ) Q/< Q + h ma ma u 3 0<v h Q/< Q 0<j v E(u, j) + O((h + hq) / log 3 ). (3.8) Now E(u, j) E(u, j) Q/< Q 0<j v 0<j v = k v τ(k) E(u, k) ( )( τ (k) k v k v E(u, k) ) v 3 4 u log 3 v log 4 u (3.9) where we have used k v τ (k) v log 3 v and Proposition in the last line. We apply (.) for the term involving I δ ( h, ) in (3.8), where the condition in (.)

11 PRIMES IN ARITHMETIC PROGRESSIONS is satisfied for h ǫ Q h, and conclude Q/< Q I(, h, ) hlog( h ) + h γ + log π + h Q + min(q 3 h log 3 Q, Qh) log p p + h 7 4 log h / log 3. (3.0) This proves Theorem. To prove the corollary, by (3.4) it is sufficient to pick Q so that the right hand side of (3.0) (or (.7)) is hqlog log. This is the case if h 3 4 log 5 Q h. If in addition, h = o() and Q = o(h/ log 3 h), then the right hand side of (3.0) is o(hq) which gives the second part of the corollary. Proposition is due to Kaczorowski, Perelli, and Pintz. The proof may be found in [7] with three modifications. First, in that paper the result is proved for the Goldbach problem, and therefore one replaces the generating function S(α) with S(α). This changes the main term to the one given in Proposition but all error terms are estimated with absolute value and therefore the rest of the proof goes through unchanged. Second, as mentioned in [8], Lemma should state /Q /Q ψ (N, χ, η) dη N log4 N Q (3.) where the original had the log factor log N instead; this slightly inflates the power of the log term in Proposition. Finally, there is an additional error term F(n, N, H) in the Kaczorowski, Perelli, and Pintz result that can be eliminated. To do this, as in [7] one derives from (3.) that a (a,)= /Q /Q R(η,, a) dη N log4 N Q (3.) and then by the Cauchy-Schwarz ineuality one obtains a (a,)= /Q /Q R(η,, a) dη N log N. (3.3) Q Using this estimate in the proof in [7] one obtains P log 3 which is smaller than the error estimate for 3 since Q. One then finds that F(n, N, H) can be absorbed into the main error term. 4. Proof of Theorem 3 The proof of Theorem 3 closely follows the proof of Theorem 4 of [] and therefore we do not repeat parts of the proof that need no alterations from the earlier proof. The information on twin primes we need is contained in the following Proposition.

12 GOLDSTON AND YILDIRIM Proposition. Assume the Generalized Riemann Hypothesis. Then we have for H R H that E(, j) H log 6. (4.) 0< j H R R< H j This generalizes Proposition 4 of [] which is the case H =. Proof of Theorem 3. Using the GRH estimate (.4) we have I(, h, ) = I(, h, ) + O(Q 0 hlog ), (4.) Q Q 0< Q where Q 0 will be chosen later. By (3.3) we therefore have on GRH that I(, h, ) = hlog( h ) h γ + log π + log p p Q Q 0< Q + O(Qh ) + I δ ( h, ) + Q 0< Q k+h ( ) k k= k Q 0< Q 0<j u k E(u, j)du + O(h / log 3 ) + O(Qh / log 3 ) + O(Q 0 hlog ) (4.3) First, an easy computation gives hlog( h ) h γ + log π + log p p Q 0< Q ( = Qhlog( Q h ) Qh γ + log π + ) log p + O(Q 0 hlog ). p(p ) p (4.4) Net by (.) we have, for h ǫ Q 0, I δ ( h, ) min(q 3 3 h log Q, Qh). (4.5) Q 0< Q Finally, by Proposition we have for h Q 0 k+h E(u, j)du k Q 0< Q = h k+h k 0<j u k ma 0 H h Q 0 R Q 0<j u k Q 0 Q 0< u k j 0<j H R R< H j E(, j) 0<j u k Q Q< u k j E(u, j)du h 3 log 6. (4.6)

13 PRIMES IN ARITHMETIC PROGRESSIONS 3 We now choose Q 0 = h. Therefore the conditions in euations (4.5) and (4.6) are satisfied, and Theorem 3 follows from (4.3), (4.4), (4.5), and (4.6). Proof of Proposition. We sketch the modifications needed in the proof of Proposition 4 in []. Since E(, k) = E(, k) we need only consider positive j. By (7.7) and (7.6) of [] we have E(, j) = I + I 3 + E + E + O(log j). (4.7) We shall denote f(j, ) = H,R j H R R< H j f(j, ). Then Lemma 7. of [] states that, for α b r r, (b, r) =, and r H R, W(α) = H,R e(jα) H log H. (4.8) r To prove Proposition we sum over j and in (4.7) and estimate each term on the right hand side. This was done in [] for the case H =. The argument is identical here only we apply the estimate (4.8) at the appropriate point in each estimation. We obtain in this way I H log 5, H,R E RH log H, H,R H,R H,R I 3 H R log6, E H R log3, log j H log. H,R (4.9) Taking H R H (which forces R / ), then all of these error terms are H log 6 and Proposition follows. 5. Proof of Theorem 4 The proof of Theorem 4 is similar to the proof of the lower bounds for I(, h) in [4] and G(, ) in []. We make use of the arithmetic function λ R (n) = r R We note the simple bound, for any ǫ > 0, We reuire the following lemma from []. µ (r) dµ(d). (5.) d r d n λ R (n) τ(n)log R n ǫ log R. (5.)

14 4 GOLDSTON AND YILDIRIM Lemma 5. Let L(R) = r R µ (r) = log R + O(). (5.3) For R, we have λ R (n)λ(n) = ψ()l(r) + O(R log ), (5.4) n λ R (n) = L(R) + O(R ). (5.5) n Letting E(;, a) = ψ(;, a) E,a φ(), where E,a = if (, a) = and is 0 otherwise, we have for 0 < k, with N, N as in (.7), that and N (k)<n N (,k) N (k)<n N (,k) ( ) kτ(k) λ R (n)λ(n + k) = S(k)( k ) + O φ(k)r + O( r R µ (r)r log(r/r) E(N + k; r, k) E(N + k; r, k) ), (5.6) ( ) kτ(k) λ R (n)λ R (n + k) = S(k)( k ) + O + O(R ). (5.7) φ(k)r Proof of Theorem 4. By (.), (.7), and (3.) we have on RH that I(, h, ) = S h h + O( φ() φ() log ). (5.8) Let a() denote a sum over a complete set of residues modulo. By (.9) and the euation above it we see that S = a() = a() (ψ(y + h;, a) ψ(y;, a)) dy + O(h( h + )log ) y<n y+h n a() Λ(n) dy + O(h( h + )log ) = S + O(h( h + )log ). (5.9) We now obtain a lower bound for S through the ineuality J(, ) = a() y<n y+h n a() (Λ(n) λ R (n)) dy 0. (5.0)

15 PRIMES IN ARITHMETIC PROGRESSIONS 5 Writing R (n) = Λ(n) λ R (n), we have by Lemma that J(, ) = ( ( R (n)) f(n,, h) a() = S <n +h n a() <n +h 0<k h k 0() + 0<k h k 0() ( <n +h k n n+k a() α(n, 0)f(n,, h) <n +h k ) ) R (n) R (n + k)f(n,, h k) α(n, k)f(n,, h k), (5.) where α(n, k) = Λ(n)λ R (n + k) + Λ(n + k)λ R (n) λ R (n)λ R (n + k). To evaluate these sums we use Lemma and Lemma 5. In order to control the error term O(R ) in (5.5) and (5.7) we assume R. (5.) By (5.3),(5.4), (5.5), (5.) and the prime number theorem we have that α(n, 0) = log R + O() n which on applying Lemma gives together with (5.) that <n +h α(n, 0)f(n,, h) = hlog R + O(h). (5.3) By replacing by + k in (5.6) and (5.7) and using the case of k negative in (5.6) to handle the term Λ(n)λ R (n + k) we obtain ( ) kτ(k) α(n, k) = S(k) + O + O(R ) φ(k)r n + O ( r R µ (r)r log(r/r) ( E( + k; r, k) + E(; r, k) + E(k; r, k) ) ). (5.4) By Lemma we now obtain α(n, k)f(n,, h k) 0<k h k 0() <n +h k = 0<j h <n +h j α(n, j)f(n,, h j) = (h j)s(j) + O(R(, h, )), (5.5)

16 6 GOLDSTON AND YILDIRIM where R(, h, ) = h R + l= By Lemma 3 we have l+h j l jτ(j) φ(j) r R + h R µ (r)r log(r/r) ma v u+j E(v; r, j) du. (5.6) (h j)s(j) = h φ() hlog h + O(h(log log 3)3, (5.7) and therefore we conclude by (5.8), (5.9), (5.0), (5.), (5.3), (5.5) and (5.7) that subject to (5.) and RH I(, h, ) hlog R h + O(h(log log 3)3 ) + O( h φ() log ) + O(R(, h, )). It remains to bound R(, h, ). Since (5.8) jτ(j) φ(j) τ() φ() jτ(j) φ(j) hǫ ( + log h ), we see where R(, h, ) h ǫ R = h log R r R R log h + h R + R, (5.9) µ (r)r ma u +h E(u; r, j). (5.0) The simplest way to bound R is to use the bound [] E(,, a) log (), which assumes GRH. This was done in [4] and []. Heath-Brown observed that one can do better by taking advantage of the averaging over j. To do this, we use Hooley s GRH estimate [6] * a() ma E(u;, u a) log 4. (5.) This result without the ma is a well known result of Turán and of Montgomery [9]. Now in the inner sum in (5.0) we insert the condition (r, j) = with an error (r,j)> ψ(3; r, j) p r n 3 p n Λ(n) h log p r h log.

17 PRIMES IN ARITHMETIC PROGRESSIONS 7 Since for (j, r) = the seuence of j will run through a reduced set of residues modulo r as j runs through a reduced set of residues modulo r, we see that R h logr r R (r,)= µ (r)r (j,r)= ma u +h By Cauchy s ineuality and (5.) we have r R (r,)= µ (r)r (j,r)= r R (r,)= ma u +h µ (r)r ( ) h r R (r,)= ( ) h ( h R r R E(u; r, j) + hr ( ) ( h (j,r)= µ ( (r)r + h r µ (r)r ) We conclude by (5.9), (5.) and (5.3) that ( ) h R R(, h, ) h log 3 We now obtain from (5.8) that + hr E(u; r, j) h R log 3 + O( ) (5.) ma u +h ) E(u; r, j) ) ( * ma E(u; r, j) j(r) u 3 ( ( ) ) h + log r ) log. (5.3) + h ǫ R log h + h R log 3 I(, h, ) hlog R h + O(h(log log )3 ) + R, + h R. (5.4) where We take ( h R R h log 3 + hr R = ( h log 3 + hǫ log R ) log 3 ) + hr. and see that R = O(h) subject to the condition that h 3. We conclude ǫ log 3 that I(, h, ) h ( ( log ) ) 3 O(h(log log ) 3 ) (5.5) h which proves Theorem 4.

18 8 GOLDSTON AND YILDIRIM References. H Davenport, Multiplicative Number Theory, second edition, revised by H. L. Montgomery, Springer-Verlag (Berlin).. J. B. Friedlander and D. A. Goldston, Variance of distribution of primes in residue classes, Quart. J. Math. Oford () 47 (996), J. B. Friedlander and D. A. Goldston, Note on a Variance in the Distribution of Primes, Conference Proceedings (997). 4. D. A. Goldston, A lower bound for the second moment of primes in short intervals, Epo. Math. 3 (995), D. A. Goldston and H. L. Montgomery, Pair correlation of zeros and primes in short intervals, Analytic Number Theory and Diophantine Problems, Birkhaüser, Boston, Mass., 987, pp C. Hooley, On the Barban-Davenport-Halberstam theorem: VI, J. London Math. Soc. () 3 (976), J. Kaczorowski, A. Perelli, and J. Pintz, A note on the eceptional set for Goldbach s problem in short intervals, Mh. Math. 6 (993), A. Languasco and A. Perelli, A pair correlation hypothesis and the eceptional set in Goldbach s problem, Mathematika 43 (996), H. L. Montgomery, Multiplicative Number Theory, Lecture Notes in Mathematics, vol. 7, Springer-Verlag, Berlin, A. E. Özlük, On the -analogue of the pair correlation conjecture, Journal of Number Theory 59 (996), K. Prachar, Generalisation of a theorem of A. Selberg on primes in short intervals, Topics in Number Theory, Debrecen, 974, pp C. Y. Yıldırım, The pair correlation of zeros of Dirichlet L-functions and primes in arithmetic progressions, Manuscripta Math. 7 (99), DAG:Department of Mathematics and Computer Science, San Jose State University, San Jose, CA 959, USA address: goldston@jupiter.sjsu.edu CYY:Department of Mathematics, Bilkent University, Ankara 06533, Turkey address: yalcin@sci.bilkent.edu.tr