aa r X i v : . [ m a t h . N T ] O c t SUMS OF TWO SQUARES IN SHORT INTERVALS
JAMES MAYNARD
Abstract.
We show that there are short intervals [ x, x + y ] containing ≫ y / numbers expressible as the sum of two squares, which is many more than theaverage when y = o ((log x ) / ). We obtain similar results for sums of twosquares in short arithmetic progressions. Introduction
Let X be large and y ∈ (0 , X ]. The prime number theorem shows that on average for x ∈ [ X, X ] we have(1) π ( x + y ) − π ( x ) ≈ y log x . The approximation (1) is known ([10], improving on [12]) to hold in the sense ofasymptotic equivalence for all x ∈ [ X, X ] provided that X / ≤ y ≤ X , whilstthe celebrated result of Maier [15] shows that (1) does not hold in the sense ofasymptotic equivalence for all x ∈ [ X, X ] when y is of size (log X ) A , for any fixed A >
0. He showed that for y = (log X ) A , there exists x ∈ [ X, X ] such that(2) π ( x + y ) − π ( x ) > (1 + δ A ) y log x for a positive constant δ A depending only on A (and he also obtained similar resultsfor intervals containing fewer than the average number of primes).It is trivial that in the range y ≪ log X , (1) cannot hold in the sense of asymptoticequivalence, and the extent to which this approximation fails is closely related to thestudy of gaps between primes. Maier and Stewart [14] combined the Erd˝os-Rankinconstruction for large gaps between primes with the work of Maier to get showthe existence of intervals containing significantly fewer primes than the average. Itfollows from the recent advances in the study of small gaps between primes ([21],[18], [16] and unpublished work of Tao) that there are also short intervals containingsignificantly more primes than the average.The situation is similar for the set S of integers representable by the sum of twosquares. Hooley [11] has shown that for almost all x ∈ [ X, X ] one has(3) S y √ log X ≪ X n ∈S∩ [ x,x + y ] ≪ S y √ log X , provided y/ √ log X → ∞ (here S > o (1)) S y/ √ log X on average over x ∈ [ X, X ]). By adapting Maier’s method, Balog and Wooley [1] showed that for any fixed
A > y = (log X ) A , there exists δ ′ A > x ∈ [ X, X ] such that(4) X n ∈S∩ [ x,x + y ] ≥ (1 + δ ′ A ) S y √ log X , and so Hooley’s result cannot be strengthened to an asymptotic which holds for all x ∈ [ X, X ] when y is of size (log x ) A .As a result of the ‘GPY sieve’, Graham Goldston Pintz and Yıldırım [6] haveshown that for any fixed m there exists x ∈ [ X, X ] such that [ x, x + O m (1)] contains m numbers which are the product of two primes ≡ m elements of S . By modifying the GPY sieve to find short intervals containing many elements of S , we show the existence of intervals [ x, x + y ] containing a higher order of magnitudethan the average number when y is of size o ((log X ) / ). Thus in this range wesee that the upper bound in Hooley’s result cannot hold for all x ∈ [ X, X ]. (It isa classical result that the lower bound cannot hold for all x in such a range; thestrongest such result is due Richards [20] who has shown there exists x ∈ [ X, X ]such that there are no integers n ∈ S in an interval [ x, x + O (log x )].)2. Intervals with many primes or sums of two squares
Let the function g : (0 , ∞ ) → R be defined by(5) g ( t ) = sup u ≥ t e γ ω ( u ) , where γ is the Euler constant and ω ( u ) is the Buchstab function . Clearly g isdecreasing, and it is known that g ( t ) > t > Theorem 2.1.
Fix ǫ > . There is a constant c ( ǫ ) > such that we have thefollowing:(1) Let X, y satisfy c ( ǫ ) ≤ y ≤ X . Then there exists at least X − / log log X values x ∈ [ X, X ] such that π ( x + y ) − π ( x ) ≥ (cid:16) g (cid:16) log y log log x (cid:17) − ǫ + c (log x )(log y ) y (cid:17) y log x . (2) Fix a ∈ N , and let Q, x satisfy c ( ǫ ) ≤ Q ≤ x/ . Then there exists at least Q − / log log Q values q ∈ [ Q, Q ] such that π ( x ; q, a ) ≥ (cid:16) g (cid:16) log x/q log log x (cid:17) − ǫ + c (log x )(log x/q ) x/q (cid:17) xφ ( q ) log x . (3) Let q, x satisfy c ( ǫ ) ≤ q ≤ x/ , and let q have no prime factors less than log x/ log log x . Then there exists at least q − / log log q integers a ∈ [1 , q ] such that π ( x ; q, a ) ≥ (cid:16) g (cid:16) log x/q log log x (cid:17) − ǫ + c (log x )(log x/q ) x/q (cid:17) xφ ( q ) log x . The Buchstab function ω ( u ) is defined by the delay-differential equation ω ( u ) = u − (0 < u ≤ , ∂∂u ( uω ( u )) = ω ( u −
1) ( u ≥ . UMS OF TWO SQUARES IN SHORT INTERVALS 3
Here c > is an absolute constant. We do not claim Theorem 2.1 is new; all the statements of Theorem 2.1 followimmediately from the work [15], [5] and [16].When the number of integers in the short interval or arithmetic progression islarge compared with log x log log x , the first terms in parentheses dominate the righthand sides, and we obtain the results of Maier [15] and Friedlander and Granville[5] that there are many intervals and arithmetic progressions which contain morethan the average number of primes by a constant factor. When the number ofintegers is small compared with log x log log x , the final terms dominate, and we seethat there are many intervals and arithmetic progressions which contain a numberof primes which is a higher order of magnitude than the average number.Part 3 of Theorem 2.1 is one of the key ingredients in recent work of the authoron large gaps between primes [17]. The previous best lower bound for the largestgaps between consecutive primes bounded by x was due to Pintz [19] who explicitlyindicated that one obstruction to improving his work was the inability of previoustechniques to show the existence of many primes in such a short arithmetic pro-gression with prime modulus (this issue was also clear in the key work of Maier andPomerance [13] on this problem). The independent improvement on this bound byFord, Green, Konyagin and Tao [3] used the Green-Tao technology to show thereare many primes q for which there were many primes in an arithmetic progres-sion modulo q , although it appears this approach does not give improvements toTheorem 2.1.The aim of this paper is to establish an analogous statement to Theorem 2.1for the set S of numbers representable as the sum of two squares. We first requiresome notation. We define the counting functions S ( x ) and S ( x ; q, a ), the constant S and the multiplicative function φ S by S ( x ) = { n ≤ x : n ∈ S} , (6) S ( x ; q, a ) = { n ≤ x : n ∈ S , n ≡ a (mod q ) } , (7) S = 1 √ Y p ≡ (cid:16) − p (cid:17) − / , (8) φ S ( p e ) = p e , p ≡ ,p e +1 / ( p + 1) , p ≡ , e − , p = 2 and e ≥ , , p e = 2 . (9)With this notation, the average number S ( x + y ) − S ( x ) of elements of S in a shortinterval [ x, x + y ] is ∼ S y/ √ log x and the average number S ( x ; q, a ) of elementsof S in the arithmetic progression a (mod q ) for ( a, q ) = 1 and a ≡ , q ))which are less than x is ∼ S x/ ( φ S ( q ) √ log x ).Finally, we let F be the sieve function occurring in the upper bound half dimen-sional sieve, that is the function defined by the delay-differential equations(10) F ( s ) = 2 e γ/ π / s / for 0 ≤ s ≤ , ( s / F ( s )) ′ = 12 s − / f ( s −
1) for s > ,f ( s ) = 0 for 0 ≤ s ≤ , ( s / f ( s )) ′ = 12 s − / F ( s −
1) for s > . JAMES MAYNARD
It is known that 1 < F ( s ) < O ( e − s ) for all s > Theorem 2.2.
Fix ǫ > , and let S denote the set of integers representable as thesum of two squares. There is a constant c ( ǫ ) > such that we have the following:(1) Let X, y satisfy c ( ǫ ) ≤ y ≤ X . Then there exists at least X − / log log X values x ∈ [ X, X ] such that S ( x + y ) − S ( x ) ≥ (cid:16) F (cid:16) log y log log x (cid:17) − ǫ + c (log x ) / y / (cid:17) S y (log x ) / . (2) Fix a ∈ S , and let Q, x satisfy c ( ǫ ) ≤ Q ≤ x/ . Then there exists at least Q − / log log Q values q ∈ [ Q, Q ] such that S ( x ; q, a ) ≥ (cid:16) F (cid:16) log x/q log log x (cid:17) − ǫ + c (log x ) / ( x/q ) / (cid:17) S xφ S ( q )(log x ) / . (3) Let q, x satisfy c ( ǫ ) ≤ q ≤ x/ , and let q have no prime factors less than log x/ log log x . Then there exists at least q − / log log q integers a ∈ [1 , q ] such that S ( x ; q, a ) ≥ (cid:16) F (cid:16) log x/q log log x (cid:17) − ǫ + c (log x ) / ( x/q ) / (cid:17) S xφ S ( q )(log x ) / . Here c > is an absolute constant. Similarly to Theorem 2.1, when the number of integers in the short interval orarithmetic progression is large compared with (log x ) / , we find that there morethan the average number of integers representable as the sum of two square, whichis a result of Balog and Wooley [1], based on Maier’s method. When the number ofintegers is small compared with (log x ) / , then the number of integers representableas the sum of two squares is of a higher order of magnitude than the average. Wenote that unlike Theorem 2.1, Theorem 2.2 improves on the Maier matrix boundsin a range beyond the trivial region by a full positive power of log x .The constant 9 /
10 appearing in the exponent in the denominators of the finalterms of Theorem 2.2 is certainly not optimal; with slightly more effort one couldcertainly improve this. We satisfy ourselves with a weaker bound for simplicityhere, the key point of interest being one can obtain a constant less than 1, andso there are intervals of length considerably larger than the average gap betweenelements of S which contain a higher order of magnitude than the average numberof elements.Analogously to the work of Granville and Soundararajan [8], we believe thephenomenon of significant failures of equidistribution in short intervals should holdin rather more general ‘arithmetic sequences’ which are strongly equidistributed inarithmetic progressions (in the sense of a weak Bombieri-Vinogradov type theorem).We hope to return to this in a future paper.3. Overview
We give a rough overview of the methods, emphasizing the similarities betweenMaier’s matrix method and the GPY method. For simplicity we concentrate on thecase when we are looking for primes in short intervals; the arguments for arithmeticprogressions are similar, and when looking for sums of two squares one wishes to‘sieve out’ only primes congruent to 3 (mod 4).
UMS OF TWO SQUARES IN SHORT INTERVALS 5
We estimate a weighted average of the number of primes in a short interval, andwish to show that this weighted average is larger than what one would obtain if theprimes were evenly distributed. More specifically, we consider the ratio(11) (cid:16) X
X One can heuristically investigate Selberg-sieve weights (12) for larger h ,and (assuming various error terms are negligble) conclude that an approximately op-timal choice of weights λ d ,...,d k should be given by λ d ,...,d k ≈ µ ( d . . . d k ) if Q ki =1 d i has all of its prime factors less than a small multiple of log x (and otherwise).This corresponds precisely to Maier’s choice of weights (13) . Admissible sets of linear functions The bounds involving the sieve function F in Theorem 2.2 follow from the ar-gument of Balog and Wooley [1], and by making minor adaptions analogous to thework of Friedlander-Granville [5]. We sketch such arguments in Section 5.Therefore the main task of this paper is to establish the bounds coming from thefinal terms in parentheses of Theorem 2.2. We will do this by an adaptation of theGPY sieve method. Indeed, we will actually prove a rather stronger result, givenby Theorem 4.1 below, which is roughly analogous to the main theorem of [16].To ease notation, we let h P i denote the set of integers composed only of primescongruent to 1 (mod 4). Similarly let h P i those composed of primes congruent to3 (mod 4). Definition ( h P i - Admissibility) . We say a set L = { L , . . . , L k } of distinct linearfunctions L i ( n ) = a i n + b i with integer coefficients is h P i -admissible if for everyprime p ≡ there is an integer n p such that ( Q ki =1 L i ( n p ) , p ) = 1 , and if a i , b i > for all i . Theorem 4.1. Let x be sufficiently large and k ≤ (log x ) / . Then there is a prime p ≫ log log x such that the following holds.Let L = { L , . . . , L k } be a h P i -admissible set of linear functions such that thecoefficients of L i ( n ) = a i n + b i satisfy < a i ≤ (log x ) / , (2 p , a i ) = 1 , and < b i < x . There exists an absolute constant C > such that { n ∈ [ x, x ] : at least k / /C of L ( n ) , . . . , L k ( n ) are in S} ≥ x exp( − (log x ) / ) . We remark that the conditions k ≤ (log X ) / , 2 ∤ a i , a i < (log X ) / and b i ≤ x are considerably weaker than what our method requires, but simplify some of thelater arguments slightly and are sufficient for our application.In the region where y , x/Q and x/q are small compared with (log x ) / , thebounds in Theorem 2.2 follow easily from Theorem 4.1. For sums of squares in shortintervals, we consider y ≤ (log x ) / ǫ . We take k = y / , and let L = { L , . . . , L k } be the h P i -admissible set of linear functions L i ( n ) = n + h i , where h , . . . , h k are the first k positive integers in h P i which are coprime with p . (This is h P i -admissible, since L i (0) is coprime to all primes p ≡ i ). We note One needs to be slightly careful about the possible effect of Siegel zeros here, but this is aminor technical issue. UMS OF TWO SQUARES IN SHORT INTERVALS 7 that h k ≪ k (log k ) / ≤ y . It then follows immediately from Theorem 4.1 thatthere are ≫ X exp( − (log X ) / ) values of n ∈ [ X, X ] such that ≫ k / = y / of the L i ( n ) are in S , and hence ≫ X − / log log X values of x ∈ [ X, X ] such that[ x, x + y ] contains at least y / elements of S .Parts 2 and 3 of Theorem 2.2 follow analogously, defining k = ( x/Q ) / and L i ( n ) = a + h i n when Q > x (log x ) − / − ǫ for part 2, and defining k = ( x/q ) / and L i ( n ) = n + h i q when q > x (log x ) − / − ǫ for part 3. We note that it followsfrom Theorem 2.2 that actually there are many intervals [ x, x + y ] containing ≫ y / (log y ) − / elements of S (rather than ≫ y / ), for the smaller range 1 ≤ y ≤ (log x ) / . 5. Irregularities from the Maier matrix method As previously mentioned, the bounds in Theorem 2.2 when y , x/Q or x/q largecompared with (log x ) / follow from minor adaptions of the argument of Balogand Wooley [1]. In particular, the relevant bound for part 2 of Theorem 2.2 followsfrom the proof of [1, Theorem 1]. We give brief outline of the argument for parts 2and 3, leaving the details to the interested reader.We first consider part 2 of Theorem 2.2. We assume that (log x ) / − ǫ ≤ x/Q ≤ (log X ) O (1) since otherwise either the result follows from Theorem 4.1, or the resultis trivial. For simplicity we shall assume that a is odd; the case for even a is entirelyanalogous. We let P = Q p ≡ ,p ≤ z p α p , where α p is the least odd integer suchthat p α p ≥ a (4 x/Q +1) and z = log x/ (log log x ) . We note that P = Q o (1 / log log Q ) .Finally, we let Q = { q ∈ [(1 − δ ) Q + 4 a, Q + 4 a ] : q ≡ a (mod 4 P ) } for some smallfixed constant δ = δ ( ǫ ) > 0. We have X q ∈Q S ( x ; q, a ) ≥ X r ≤ x/Q − X q ∈Q S ( a + rq )= X r ≤ x/Q − X m ∈ [(1 − δ ) Q/ P,Q/ P ] S ( a (4 r + 1) + 4 mrP ) . (16)The inner sum is counting elements of S in an arithmetic progression to modulus4 rP . We note that by construction of P we must have ( P, a ) = e for some e ∈ h P i since a ∈ S , and similarly if the inner sum is non-empty we must have ( P, a (4 r +1)) = e d since a (4 r + 1) + 4 mrP ∈ S . Moreover P/ ( e d ) and P are composed ofthe same prime factors, since each prime occurs in P with odd multiplicity. Thus,if r is such that ( P, r + 1) = d , the inner sum is(17) (1 + o (1)) S δrQ/ ( e d ) φ S (4 rP/ ( d e )) p log rQ/d e ∼ S δQ φ S ( P ) √ log x . Therefore, letting ud = 4 r + 1, we obtain(18) X q ∈Q S ( x ; q, a ) ≥ (1 + o (1)) S δQ φ S ( P ) √ log x X d | P X u< (4 x/Q +1) /d u ≡ u,P )=1 . JAMES MAYNARD This double sum is exactly the sum R + which is estimated in [1, proof of Lemma4.3]. In particular, they show that(19) X d | P X u< (4 x/Q +1) /d u ≡ u,P )=1 ∼ φ S ( P ) P F (cid:16) log x/Q log z (cid:17) . Recalling that z = log x/ (log log x ) , one arrives at (for δ sufficiently small)(20) X q ∈Q S ( x ; q, a ) − (cid:16) F (cid:16) log x/q log log x (cid:17) − ǫ (cid:17) S xφ S ( q ) √ log x ! ≫ xP √ log x . This implies the relevant bound in part 2 of Theorem 2.2.We now consider part 3 of Theorem 2.2. We let ˜ P = Q p ≡ ,p ≤ z p β p where β p is the least odd integer such that p β p ≥ x/q , and let A = { a ∈ [(1 − δ ) q, q ] : a ≡ a (mod P ) } . We see that X a ∈A S ( x ; q, a ) ≥ X r ≤ ( x − q ) /q X a ∈A S ( a + rq )= X r ≤ x/q − X m ∈ [((1 − δ ) q − a ) /P, ( q − a ) /P ] S ( a + rq + mP ) . (21)We see the inner sum counts elements of S in an arithmetic progression modulo P . We choose a ≡ q/ P ), so that ( a + rq, P ) = (4 r + 1 , P ) since, byassumption, q has no prime factors in common with P . Thus (4 r + 1 , P ) = d forsome d ∈ h P i by the same argument as above, and we obtain X a ∈A S ( x ; q, a ) ≥ (1 + o (1)) S δqφ S ( P ) √ log x X d | P X u< (4 x/q − /d u ≡ u,P )=1 . (22)Again, using (19), we obtain(23) X a ∈A S ( x ; q, a ) − (cid:16) F (cid:16) log x/q log log x (cid:17) − ǫ (cid:17) S xφ S ( q ) √ log x ! ≫ xP √ log x , which implies the relevant bound for part 3.6. Setup for Theorem 4.1 The improvements of the GPY sieve in the author’s work [18, 16] are not sig-nificant for the application of finding sums of two squares in short intervals (theimprovement would be y o (1) in the final term for part 1 of Theorem 2.2, for exam-ple), and so we will use a uniform version of the simplest GPY sieve.In order to get a result that applies in the larger range, however, it is necessaryto modify the sieve to the application; a uniform version of the argument in [6] (i.e.counting numbers with two prime factors both congruent to 1 (mod 4)) would onlygive non-trivial results in the region y ≪ (log x ) / log log x , for example.We follow a similar argument to [6] (and attempt to keep the notation similar),but make modifications to allow for uniformity (similar to those in [7] and [16],although it is simpler in this context) and to specialize so as to remove only primes UMS OF TWO SQUARES IN SHORT INTERVALS 9 congruent to 3 (mod 4) (so we will apply sieves of ‘dimension’ k/ k + 1) / k and k + 1).In order to state our setup, we require the following Lemma. Lemma 6.1. Let x > and ǫ > . There exists a prime p ∈ [(log log x ) / , x ] such that X q Lemma 7.1. Assume the hypotheses of Theorem 4.1 and that sup d | λ d | ≪ (log X ) k/ .Then X X From the definition of w n , we have(30) X X Lemma 7.2. Assume the hypotheses of Theorem 4.1 and that sup d | λ d | ≪ (log X ) k/ .Then X X 1) different sumsof primes in arithmetic progressions to modulus 4 a j W Q p | de p (since 2 ∤ a j ). Thusthe inner sum is 1 φ (4 a j W ) (cid:16) π (cid:16) a j X + b j m (cid:17) − π (cid:16) a j X + b j m (cid:17)(cid:17) Y p | de ν ( p ) − p − O (cid:16) E (cid:16) a j Xm ; 4 a j W [ d, e ] (cid:17) Y p | de k (cid:17) , (36) where(37) E ( x ; q ) = sup ( a,q )=1 x ′ ≤ x (cid:12)(cid:12)(cid:12) π ( x ′ ; q, a ) − π ( x ′ ) φ ( q ) (cid:12)(cid:12)(cid:12) , [ d, e ] = Y p | de p. We first consider the error term of (36). Letting r = 4 a j W [ d, e ] ≪ X / (notingthat ( r, p ) = 1) and λ max = sup d | λ d | , this contributes at most kλ X m Fix A > . Let /A ≤ κ ≤ (log R ) / and let f : [0 , → R be asmooth non-negative function. Let γ ( p ) satisfy ≤ γ ( p ) ≤ min( Aκ, (1 − /A ) p ) forall p < R , and − L ≤ X w
For f = 1, this is a version of [9, Lemma 5.4], with the dependence on κ kept explicit. In particular, this is obtained in the proof of [4, Lemma 5.1]. Thecase of f = 1 then follows immediately by partial summation. (cid:3) Lemma 7.4. Let k ≤ (log X ) / , and let λ d be defined by λ d = µ ( d ) (cid:16)Y p | d pν ( p ) (cid:17) X d | r y r (cid:16)Y p | r ν ( p ) p − ν ( p ) (cid:17) , for ( d, p W ) = 1 , where y r = ( µ ( r ) , if r ∈ h P i , ( r, W p ) = 1 , r < X / , , otherwise.Then X d,e λ d λ e Y p | de ν ( p ) − p − ≫ k − / (log X ) / φ ( W ) W X d,e λ d λ e Y p | de ν ( p ) p ≫ . Proof. We first note that the y r variables diagonalize the second quadratic form.Substituting the expression for λ d in terms of y r given by then Lemma, we obtain X d,e λ d λ e Y p | de ν ( p ) p = X r,s y r y s (cid:16)Y p | r ν ( p ) p − ν ( p ) (cid:17)(cid:16)Y p | s ν ( p ) p − ν ( p ) (cid:17) × X d,e : d | r,e | s µ ( d ) µ ( e ) (cid:16)Y p | d pν ( p ) (cid:17)(cid:16)Y p | e pν ( p ) (cid:17)(cid:16)Y p | de ν ( p ) p (cid:17) . (42)The sum over d, e vanishes unless r = s , and so we find that X d,e λ d λ e Y p | de ν ( p ) p = X r y r Y p | r ν ( p ) p − ν ( p ) . (43)We now introduce new variables y ∗ r to perform an analogous diagonalization for thefirst quadratic form. Let y ∗ r be zero unless r ∈ h P i , ( r, W p ) = 1, r < X / and ν ( p ) > p | r . In this case, let y ∗ r be given by(44) y ∗ r = µ ( r ) (cid:16)Y p | r p − ν ( p ) ν ( p ) − (cid:17) X d : r | d λ d (cid:16)Y p | d ν ( p ) − p − (cid:17) . We see from this that for any d satisfying the same conditions, we have µ ( d ) λ d = (cid:16)Y p | d p − ν ( p ) − (cid:17) X e : d | e λ e (cid:16)Y p | e ν ( p ) − p − (cid:17) X r : d | r, r | e µ ( r )= (cid:16)Y p | d p − ν ( p ) − (cid:17) X r : d | r y ∗ r (cid:16)Y p | r ν ( p ) − p − ν ( p ) (cid:17) . (45)This gives λ d in terms of y ∗ r . Performing the analogous computation to (43) with y ∗ r in place of y r , we obtain(46) X d,e λ d λ e Y p | de ν ( p ) − p − X r ( y ∗ r ) Y p | r ν ( p ) − p − ν ( p ) . Substituting the expression for λ d in terms of y r from the statement of the lemmain (44) gives (for r such that y ∗ r = 0) y ∗ r = µ ( r ) (cid:16)Y p | r p − ν ( p ) ν ( p ) − (cid:17) X s : r | s y s (cid:16)Y p | s ν ( p ) p − ν ( p ) (cid:17) X d : r | d, d | s µ ( d ) (cid:16)Y p | d p ( ν ( p ) − ν ( p )( p − (cid:17) = µ ( r ) r X s : r | s y s φ ( s ) . (47)We can now use Lemma 7.3 and definition of y r from the statement of the lemmato evaluate this sum. We take γ ( p ) = 1 if p ≡ p, rp W ) = 1, andtake γ ( p ) = 0 otherwise. We see that for any 2 ≤ w ≤ z ≤ X / we have(48) − X p | rp W log pp + O (1) ≤ X w 12 log z/w ≪ . The sum on the left hand side is O (log log X ). Thus, by applying Lemma 7.3 tothe sum (47), we have for squarefree r < X / with r ∈ h P i and ( r, W p ) = 1and Q p | r ( ν ( p ) − > y ∗ r = rφ ( r ) X t Thus, we obtain (with κ = k/ L ≪ k log log X ) X r 1) for p ≡ p, W p ) = 1, and take γ ( p ) = 0 otherwise. We obtain (with κ = ( k − / L ≪ k log log X ) X r Let k ≤ (log X ) / and λ d be as given by Lemma 7.4. Then | λ d | ≪ (log X ) k/ . Proof. This is an immediate application of Lemma 7.3. We take γ ( p ) = ν ( p ) if p ≡ p, dW p ) = 1, and γ ( p ) = 0 otherwise. By Lemma 7.3 (taking κ = k/ L ≪ k log log X ) we have | λ d | = (cid:16)Y p | d pp − ν ( p ) (cid:17) X r 2) exp (cid:16) k X (log X ) /
By Lemmas 7.4 and 7.5, there is a choice of coefficients λ d such that | λ d | ≪ (log X ) k/ and X d,e λ d λ e Y p | de ν ( p ) − p − ≫ k − / (log X ) / φ ( W ) W X d,e λ d λ e Y p | de ν ( p ) p ≫ . Thus, by Lemmas 7.1 and 7.2, this choice of λ d corresponds to a choice of w n ≥ X X Lemma 7.5 shows | λ d | ≪ (log X ) k/ , so we have(60) w n ≪ (log X ) k (cid:16) X d | Q L i ( n ) λ d =0 (cid:17) . UMS OF TWO SQUARES IN SHORT INTERVALS 17 Thus X X ≤ n ≤ X w n ≪ (log X ) k X d ,d ,d ,d The work in this paper was started whilst the author was a CRM-ISM Postdoc-toral Fellow and the Universit´e de Montr´eal, and was completed whilst he was aFellow by Examination at Magdalen College, Oxford. References [1] Antal Balog and Trevor D. Wooley. Sums of two squares in short intervals. Canad. J. Math. ,52(4):673–694, 2000.[2] H. Davenport. Multiplicative Number Theory . Graduate Texts in Mathematics. Springer NewYork, 2000.[3] K. Ford, B. Green, S. Konyagin, and T. Tao. Large gaps between consecutive primes. preprint, http://arxiv.org/abs/1408.4505 .[4] Kevin Ford, Sergei V. Konyagin, and Florian Luca. Prime chains and Pratt trees. Geom.Funct. Anal. , 20(5):1231–1258, 2010.[5] John Friedlander and Andrew Granville. Limitations to the equi-distribution of primes. III. Compositio Math. , 81(1):19–32, 1992.[6] D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yıldırım. Small gaps between productsof two primes. Proc. Lond. Math. Soc. (3) , 98(3):741–774, 2009.[7] Daniel A. Goldston, J´anos Pintz, and Cem Yal¸cin Yıldırım. Primes in tuples. II. Acta Math. ,204(1):1–47, 2010. [8] Andrew Granville and K. Soundararajan. An uncertainty principle for arithmetic sequences. Ann. of Math. (2) , 165(2):593–635, 2007.[9] H. Halberstam and H.E. Richert. Sieve methods . L.M.S. monographs. Academic Press, 1974.[10] D. R. Heath-Brown. The number of primes in a short interval. J. Reine Angew. Math. ,389:22–63, 1988.[11] Christopher Hooley. On the intervals between numbers that are sums of two squares. IV. J.Reine Angew. Math. , 452:79–109, 1994.[12] M. N. Huxley. On the difference between consecutive primes. Invent. Math. , 15:164–170, 1972.[13] H. Maier and C. Pomerance. Unusually large gaps between consecutive primes. Trans. Amer.Math. Soc. , 322(1):201–237, 1990.[14] H. Maier and C. L. Stewart. On intervals with few prime numbers. J. Reine Angew. Math. ,608:183–199, 2007.[15] Helmut Maier. Primes in short intervals. Michigan Math. J. , 32(2):221–225, 1985.[16] J. Maynard. Dense clusters of primes in subsets. preprint, http://arxiv.org/abs/1405.2593 .[17] J. Maynard. Large gaps between primes. preprint, http://arxiv.org/abs/1408.5110 .[18] J. Maynard. Small gaps between primes. Ann. of Math.(2), to appear .[19] J. Pintz. Very large gaps between consecutive primes. J. Number Theory , 63(2):286–301,1997.[20] Ian Richards. On the gaps between numbers which are sums of two squares. Adv. in Math. ,46(1):1–2, 1982.[21] Y. Zhang. Bounded gaps between primes. Ann. of Math. (2) , 179(3):1121–1174, 2014. E-mail address ::