aa r X i v : . [ m a t h . N T ] F e b CORRELATIONS OF ALMOST PRIMES
NATALIE EVANS
Abstract.
Let p , . . . , p be primes with p p , p ∈ ( X, X ], let h be a non-zero integer and A > p , p lie in a certain range depending on X , we prove an asymptotic for the weighted numberof solutions to p p = p p + h which holds for almost all 0 < | h | ≤ H with log ε X ≤ H ≤ X log − A X .Using the same methods and supposing instead that p p , p p have the typical factorisation we prove anasymptotic which holds for almost all 0 < | h | ≤ H with exp (cid:0) (log X ) − o (1) (cid:1) ≤ H ≤ X log − A X and anasymptotic for the weighted number of solutions to p p = p + h which holds for almost all 0 < | h | ≤ H with X / ε ≤ H ≤ X log − B X and B > Introduction
We are interested in counting with suitable weights the number of solutions to the equation p p = p p + h “on average” over the shift h , where p , p , p , p are primes with p p ∈ ( X, X ], p , p of predeterminedsize in terms of X and h a non-zero integer.First considering similar questions on primes, the generalised twin prime conjecture states that for anyinteger k ≥ p such that p := p +2 k is also a prime. To get an asymptoticfor the weighted number of solutions to p = p + h with p ∈ ( X, X ] and h a non-zero integer, we canstudy the correlation X X A > 0. Matom¨aki, Radziwi l l and Tao [19] improved this range, showing that if 0 ≤ h ≤ X − ε and X / ε ≤ H ≤ X − ε then (1.2) holds for all but O ε,A ( H log − A X ) values of h such that | h − h | ≤ H .These results are proved using the Hardy-Littlewood circle method.Chen’s theorem gives that p + 2 = q such that p is prime and q is either a prime or a product of twoprimes holds for infinitely many primes p . Debouzy [3] proved under the Elliott-Halberstam conjecture thatgiven any 0 ≤ β < γ there exists X such that for all X ≥ X we have that X n ≤ X Λ( n )Λ( n + 2) + 1 γ − β X n ≤ X Λ( n + 2) X d d = nn β ≤ d ≤ n γ Λ( d )Λ( d )log n = 2Π X (1 + o (1)) . This result is proved using an improvement of the Bombieri asymptotic sieve. The Elliott-Halberstamconjecture [4] (see also [2, 5]) concerns the distribution of primes in arithmetic progressions and states thatfor every A > < θ < X q ≤ x θ max ( a,q )=1 (cid:12)(cid:12)(cid:12)(cid:12) ψ ( x ; q, a ) − xϕ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ x log A x , where we define ψ ( x ; q, a ) := P n ≤ x,n ≡ a ( q ) Λ( n ).Bombieri [1] instead considered pairs P k and P k + 2 = p with p prime and P k an almost prime with atmost k factors. More precisely, defining Λ k ( n ) := ( µ ∗ log k )( n ) to be the generalised von Mangoldt functionwhere ∗ denotes Dirichlet convolution, Bombieri proved that if k ≥ x ≥ x ( k ) we have X n ≤ X Λ k ( n )Λ( n + 2) = 2Π X (log X ) k − ( k + O ( k / − k/ ))and, assuming the Elliott-Halberstam conjecture, for k ≥ X n ≤ X Λ k ( n )Λ( n + 2) ∼ kX (log X ) k − . There are a number of results regarding bounded gaps between the primes; Zhang [29] proved thatlim inf n →∞ ( p n +1 − p n ) < × (1.3)and in particular that there exist infinitely many bounded gaps between the primes. Maynard [21] improvedthe above bound to 600, while the Polymath 8b [25] project subsequently improved this to 246. Under thegeneralised Elliott-Halberstam conjecture, the best known bound is 6. The twin prime conjecture wouldamount to proving the above result with the bound 2. Goldston, Graham, Pintz and Yıldırım [8] proved analmost prime analogue of (1.3); if q < q < · · · denotes the sequence of products of exactly two distinctprimes, then lim inf n →∞ ( q n +1 − q n ) ≤ . Returning to correlations of almost primes, we aim to prove a result analogous to (1.2) in the case ofalmost primes which are products of exactly two primes using the Hardy-Littlewood circle method. Weremark that, in the case of correlations of products of exactly two primes, due to the parity problem we donot currently know how to apply sieve methods to prove an asymptotic for a fixed shift h , even assumingthe Elliott-Halberstam conjecture. We now define the following weighted counting function: Definition 1.1. Let P , P ≥ f : N → R to be f ( n ) = ( log p , if n = p p with P < p ≤ P , , otherwise . Mikawa proved his result (in his notation) also in the range X − ε ≤ H ≤ X . Matom¨aki, Radziwi l l and Tao note that theirresult can also be proved in this range by their methods. ORRELATIONS OF ALMOST PRIMES 3 As with the work on primes discussed above, in order to prove an asymptotic for the weighted number ofsolutions “on average” over the shift h to p p = p p + h with p p ∈ ( X, X ] and p , p the size of a powerof log X , we wish to prove an asymptotic for the correlation X X 0. In fact, weprove an asymptotic which holds for H as small as log ε X : Theorem 1.2. Let ε > , A > and let log ε X ≤ H ≤ X log − A X . Define f as in Definition 1.1, let δ > be sufficiently small and define P := log B X , P := P δ with B = ( 17 + ε, if log ε X ≤ H ≤ exp((log X ) o (1) ) , + ε, if exp((log X ) o (1) ) < H ≤ X log − A X. Then, there exists some η = η ( ε ) > such that for all but at most O ( H log − η X ) values of < | h | ≤ H wehave that X X The range X log − A X ≤ H ≤ X can also be dealt with by the same methods, see for example[22], [19]. The smallest possible choice of H in the above is H = log ε X , however it may be possible tolower this exponent. In the proof of Theorem 1.2 we apply the argument of Ter¨av¨ainen [26, Sections 2-4]showing that almost all intervals [ x, x + log ε x ] contain a product of exactly two primes. The second halfof Ter¨av¨ainen’s paper is dedicated to lowering the exponent 5 + ε to 3 . 51 through an argument additionallyusing some sieve theory and the theory of exponent pairs. We do not apply these ideas here, but it is possiblethat adapting some aspects of this argument to our proof could lower the exponent of H . Remark 1.4. As H becomes an arbitrarily large power of log X , or is larger than any power of log X , weare able to improve the bound on the error terms to O ( X log − A X ) for A > H , P and the parameters of the circle method.By Mertens’ theorem, almost all products of exactly two primes p p ≤ X with p ≤ p satisfy p ∈ h exp (cid:16) (log X ) ε ( X ) (cid:17) , exp (cid:16) (log X ) − ε ( X ) (cid:17)i , (1.7)where ε ( X ) = o (1). Taking this interval in place of the restriction p ∈ ( P , P δ ] in Theorem 1.2 andmaking some adjustments to the proof, we are able to obtain an asymptotic for the correlation (1.4) whichholds for almost all 0 < | h | ≤ H , at the cost of taking H larger. Theorem 1.5. Let B > , A > and let exp (cid:0) (log X ) − o (1) (cid:1) ≤ H ≤ X log − A X . Define f as in Defini-tion 1.1 with P := exp (cid:0) (log X ) ε ( X ) (cid:1) and P := exp (cid:0) (log X ) − ε ( X ) (cid:1) , where ε ( X ) = o (1) . Then, for all butat most O ( H log − B X ) values of < | h | ≤ H we have that X X We can also combine this argument with the work of Mikawa [22] on correlations of the von Mangoldtfunction to prove an asymptotic for the weighted number of solutions to p p = p + h where p p factorisesin the typical way (1.7) which holds for almost all 0 < | h | ≤ H with H as small as X / ε . Theorem 1.6. Let ε > be sufficiently small, B > , A > and let X / ε ≤ H ≤ X log − A X . Define f as in Definition 1.1 with P , P defined as in Theorem 1.5. Then, for all but at most O ( H log − B X ) valuesof < | h | ≤ H we have that X X As in previous works on correlations of the von Mangoldt function, for theproof of Theorem 1.2 we apply the Hardy-Littlewood circle method, expressing the correlation (1.4) in termsof the integral Z | S ( α ) | e ( − hα ) dα, (1.8)where S ( α ) is the exponential sum S ( α ) = X X 1) such that | α − a/q | ≤ / ( q log C X ) for some integers ( a, q ) = 1 with 1 ≤ q ≤ log A ′ X for some 0 < A ′ < C , and theminor arcs consisting of the rest of the circle.In many problems of this type (see e.g. [19], [22]) where the Hardy-Littlewood circle method is applied,it is usual that the major arcs are treated in a standard way to provide the main term and an error termwhich is not too difficult to control, while the contribution from the minor arcs is more difficult to boundsuitably. Since the correlation X X ORRELATIONS OF ALMOST PRIMES 5 The major arcs contribute the main term, which is evaluated in a standard way, and an error term. Weexpand the exponential sum (1.9) in terms of Dirichlet characters, with a suitable approximation to theprincipal character providing the main term.To the remaining terms in the expansion of S ( α ), we again apply Gallagher’s Lemma to reduce theproblem to understanding almost primes in almost all short intervals. We add and subtract a sum over alonger interval, so that we aim to estimate an expression of the form X q ≤ log A ′ X qϕ ( q ) X χ ( q ) χ = χ Z XX (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q log C X X x Throughout p, p i , are used to denote prime numbers, while k, l, m, n, q, r, v (with or withoutsubscripts) are positive integers.As usual, µ ( · ) is the M¨obius function and ϕ ( · ) is the Euler totient function. We let d r ( n ) denote thenumber of solutions to n = a · · · a r in positive integers. We let c q ( · ) be the Ramanujan sum, defined by c q ( n ) := q X a =1( a,q )=1 e (cid:18) anq (cid:19) . NATALIE EVANS We write τ ( · ) for the Gauss sum defined on Dirichlet characters χ modulo q by τ ( χ ) := q X n =1 e (cid:18) nq (cid:19) χ ( n ) , (1.11)which satisfies τ ( χ ) = µ ( q ).We use e : T → R to denote e ( x ) := e πix , where T is the unit circle. The notation S ( · ) is theindicator function of the set S ; in particular, we write S ( n ) = 1 if n ∈ S and S ( n ) = 0 otherwise. Let k x k := min n ∈ Z | x − n | denote distance to the nearest integer.We will use ( a, b ) to denote the greatest common divisor of natural numbers a and b , while we write a | b if a divides b . The shorthand a ≡ b ( q ) is used to denote that a and b are congruent modulo q .We use the shorthand χ ( q ) to denote that the summation is taken over all Dirichlet characters modulo q . For complex functions g , g we use the usual asymptotic notation g ( x ) = O ( g ( x )) or g ( x ) ≪ g ( x ) todenote that there exist real x and C > x ≥ x we have that | g ( x ) | ≤ C | g ( x ) | . Wewrite g ( x ) = o ( g ( x )) if for every ε > x such that | g ( x ) | ≤ ε | g ( x ) | for all x ≥ x . We usethe convention that ε > Preliminary Lemmas We now state several results we will need throughout the argument. We will need the following bound onprimes p such that p + h is also prime and the singular series: Lemma 2.1. Let h be an even non-zero integer and suppose that y ≥ . The number of primes p ∈ ( x, x + y ] such that p + h is also prime is ≪ S ( h ) y (log y ) . Furthermore, we have that X h ≤ x S ( h ) ≪ x. Proof. See [24, Corollary 3.14] and the subsequent exercises. (cid:3) We will also need Gallagher’s Lemma, which will reduce bounding integrals over the major and minorarcs to studying almost primes in short intervals. Lemma 2.2 (Gallagher’s Lemma) . Let < y < X/ . For arbitrary complex numbers a n , we have Z | β |≤ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X X This Lemma is a modification of [7, Lemma 1] (see also [22, Lemma 1]). (cid:3) Once we have applied Gallagher’s Lemma in the treatment of the major arcs, part of the error term isreduced to a Dirichlet character analogue of a problem on primes in almost all short intervals. We willuse the following result adapted from the work of Koukoulopoulos [17] to bound the second term arising in(1.10): Lemma 2.3. Let A ≥ and ε ∈ (0 , ] . Let X ≥ , ≤ Q ≤ ∆ X / ε and ∆ = X θ with + 2 ε ≤ θ ≤ .Then we have that X q ≤ Q X χ ( q ) Z XX (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ORRELATIONS OF ALMOST PRIMES 7 We use the following Parseval-type result to reduce the problem of finding almost primes in short intervals(cf. the first term of (1.10)) to finding cancellation in the mean square of the associated Dirichlet polynomial: Lemma 2.4 (Parseval Bound) . Let a n be arbitrary complex numbers, and let ≤ h ≤ h ≤ XT with T ≥ .Define F ( s ) := P X This is [26, Lemma 1], which is a variant of [18, Lemma 14]. (cid:3) Finally, we record an exponential sum bound and a related bound on the sum of reciprocal of the distanceto the nearest integer function which provide the necessary cancellation in the estimation of the minor arcs. Lemma 2.5. Let β ∈ R , then X n ≤ x e ( βn ) ≪ min (cid:18) x, k β k (cid:19) . Proof. This is a standard result, see for example [14, Chapter 8]. (cid:3) Lemma 2.6. If < X ≤ Y and α ∈ R satisfies α = a/q + O ( q − ) with ( a, q ) = 1 , then we have X n ≤ X min (cid:18) Yn , k αn k (cid:19) ≪ (cid:18) Yq + X + q (cid:19) log( qX ) . Proof. This is a standard result, see for example [14, Chapter 13]. (cid:3) Applying the Circle Method To prove Theorem 1.2, we will apply the Hardy-Littlewood circle method. Let ε > P := log B X . We define B > H as follows: B := ( 17 + ε, if log ε X ≤ H ≤ exp((log X ) o (1) ) , + ε, if exp((log X ) o (1) ) < H ≤ X log − A X (3.1)and we define P := P δ with δ > Bδ is sufficiently small in terms of ε ). Forsimplicity we will denote P := P from now on. Defining f as in Definition 1.1 with this choice of P , weconsider the integral Z | S ( α ) | e ( − hα ) dα = X X 1) we define the exponential sum S ( α ) := X X 1) such that (cid:12)(cid:12)(cid:12)(cid:12) α − aq (cid:12)(cid:12)(cid:12)(cid:12) ≤ qQ for some 1 ≤ q ≤ Q , a < q, ( a, q ) = 1 (3.5)with Q := log A ′ X and Q := P log X . Here we define A ′ > H as follows A ′ := ( ε , if log ε X ≤ H ≤ exp((log X ) o (1) ) , ε , if exp((log X ) o (1) ) < H ≤ X log − A X. (3.6)We define the minor arcs m to be the rest of the circle, that is, the set of real α ∈ (0 , 1) such that (cid:12)(cid:12)(cid:12)(cid:12) α − aq (cid:12)(cid:12)(cid:12)(cid:12) ≤ qQ for some Q < q ≤ Q, a < q, ( a, q ) = 1 . (3.7) Remark 3.1. The parameters satisfy Q < P < Q < H . Decreasing the size we can take P would directlyreduce how small we are able to take H .In Section 4, we will prove the following estimate for the integral over the minor arcs: Proposition 3.2 (Minor Arc Estimate) . Let A > and let ε > be sufficiently small. Let Q log ε X ≤ H ≤ X log − A X . With m defined as in (3.7) , for α ∈ m there exists some η = η ( ε ) > such that Z m ∩ [ α − H ,α + H ] | S ( θ ) | dθ ≪ X log η X . (3.8)Sections 5 to 8 will be dedicated to proving the following expression for the integral over the major arcs: Proposition 3.3 (Major Arc Integral) . Let A > and let ε > be sufficiently small. Let log ε X ≤ H ≤ X log − A X . With M defined as in (3.5) and δ > sufficiently small, there exists some η = η ( ε ) > suchthat for all but at most O ( HQ − / ) values of < | h | ≤ H we have that Z M | S ( α ) | e ( − hα ) dα = S ( h ) X X P
Proof of Theorem 1.2. Following [19, Section 3], by [19, Proposition 3.1] we have that X < | h |≤ H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X X ORRELATIONS OF ALMOST PRIMES 9 Noting that by partial summation and Mertens’ theorem we have the bound Z | S ( α ) | dα = X X Applying Chebyshev’s inequality and Proposition 3.3 then gives the result. (cid:3) The Minor Arcs We first treat the integral over the minor arcs, proving Proposition 3.2 by following the proof of [22,Lemma 8]. Proof of Proposition 3.2. Starting with the minor arc integral (3.8), we make the substitution θ = α + β tosee that I := Z m ∩ [ α − H ,α + H ] | S ( θ ) | dθ = Z α + β ∈ m | β |≤ H | S ( α + β ) | dβ. We apply Lemma 2.2 to the integral to get I ≪ H Z XX (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x 2, thecondition 0 ≤ x ≤ X is weaker than the condition max( mp , mp ) − H ≤ x < min( mp , mp ). Therefore,if m | p − p | ≤ H we have that | Ω | = H − m | p − p | .We now split the sum into the diagonal terms, p = p , and the off-diagonal terms, p = p , denoted by S and S respectively. The diagonal terms contribute S ≪ P δ H log P X P Now we bound the off-diagonal terms. Let r = | p − p | . Noting that 0 < mr ≤ H , we need to bound S ≪ P δ H log P X P < m ≤ P δ , we have that 0 < r ≤ H/P . We apply partial summationand Lemma 2.5 to the sum over m to see that S ≪ P δ H log P X Therefore the contribution of the off-diagonal terms can be bounded by S ≪ XP δ HP log P X 0. Otherwise, if H > exp((log X ) o (1) ), we have that log QHP ≪ log X and Q =log ε X , so that I ≪ XP δ (cid:18) log X (cid:18) Q + 1 P + QH (cid:19) + P log( X/P ) H (cid:19) ≪ X log η X for some η = η ( ε ) > 0, which is acceptable. (cid:3) The Major Arcs We now shift our attention to evaluating the contribution of the integral over the major arcs. We will firstexpand the exponential sum S ( α ) in terms of Dirichlet characters and suitably approximate the contributionof the principal character, which will provide the main term. We will then evaluate this main term and thesequel will then be dedicated to bounding the error terms that arise from this expansion. ORRELATIONS OF ALMOST PRIMES 11 Expanding the Exponential Sum. First, we rewrite the integral over the major arcs by expandingthe exponential sum S ( α ) in terms of Dirichlet characters. We first define a ( α ) := µ ( q ) ϕ ( q ) X P
Let M be defined as in (3.5) . We have that Z M | S ( α ) | e ( − hα ) dα = Z M | a ( α ) | e ( − hα ) dα + O (cid:0) A ( X ) B ( X ) + B ( X ) (cid:1) . Proof. Let α ∈ M , so that α = aq + β with q ≤ Q , ( a, q ) = 1 and | β | ≤ qQ . Then S ( α ) = X X In this section we evaluate the integral R M | a ( α ) | e ( − hα ) dα , giving themain term of the asymptotic (and a bound for A ( X )): Proposition 5.2. Let ε > be sufficiently small. Then for all but at most O ( HQ − / ) values of < | h | ≤ H we have that Z M | a ( α ) | e ( − hα ) dα = S ( h ) X X P , where we define the singular series S ( h ) as in Theorem 1.2. Before we can prove Proposition 5.2, we need an expression involving the singular series S ( h ). Lemma 5.3 (The Singular Series) . Let h be a non-zero even integer and Q be defined as in (3.6) . Then,for all but at most O ( HQ − / ) values of < | h | ≤ H we have that X q ≤ Q µ ( q ) c q ( − h ) ϕ ( q ) = S ( h ) + O ( Q − / log H ) . Proof. For similar results, see [19, Page 39] and [27, Page 35]. Rewriting the sum over q , we have that X q ≤ Q µ ( q ) c q ( − h ) ϕ ( q ) = ∞ X q =1 − X q>Q µ ( q ) c q ( − h ) ϕ ( q ) . The first term can be seen to be equal to S ( h ) by calculating the Euler product. It remains to bound thetail of the sum. By [27, Page 35], we have that X By Chebyshev’s inequality, we have for all but at most O ( HQ − / ) values of h the bound X q>Q µ ( q ) c q ( − h ) ϕ ( q ) ≪ Q − / log H, as claimed. (cid:3) We are now able to complete the proof of Proposition 5.2. Proof of Proposition 5.2. Applying the definition of the major arcs (3.5) and expanding the square, we havethat Z M | a ( α ) | e ( − hα ) dα = X q ≤ Q X ≤ a ≤ q ( a,q )=1 Z | β |≤ qQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ ( q ) ϕ ( q ) X P 0, as claimed. (cid:3) The Error Term of the Major Arcs In order to complete the proof of Proposition 3.3, and therefore the proof of Theorem 1.2, we need tofind sufficient cancellation in the error term B ( X ) arising on the major arcs. In this section we prove thefollowing bound for B ( X ), which immediately completes the proof of Proposition 3.3 when combined withProposition 5.2: Proposition 6.1. Let ε > be sufficiently small, then there exists some η = η ( ε ) > such that B ( X ) ≪ X log η X . Reduction of the problem. First, using Gallagher’s Lemma (Lemma 2.2), we will reduce the problemof estimating B ( X ) to understanding almost primes in almost all short intervals. We first define thefollowing: let ∆ := X/T with T := X / and B ( X ) := X q ≤ Q qϕ ( q ) X χ ( q ) Z XX (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) qQ X x We have that B ( X ) ≪ B ( X ) + B ( X ) . Then, if we can prove that B i ( X ) ≪ X log − η X for i = 1 , 2, we will immediately be able to concludeProposition 6.1. Proof. By definition, we have that B ( X ) = X q ≤ Q X ≤ a ≤ q ( a,q )=1 Z | β |≤ qQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ( q ) X χ ( q ) τ ( χ ) χ ( a ) X X Let C > , then with B ( X ) as defined in (6.2) we have B ( X ) ≪ X log C X . Proof. We separate the cases χ = χ and χ = χ . If χ = χ , we have that2 q ∆ X x 0, as required. (cid:3) Bounding B ( X ) . It now remains to prove the required bound for B ( X ). This problem can bereduced to finding cancellation in the mean square of a Dirichlet polynomial. Proposition 6.4. Let ε > be sufficiently small. With B ( X ) as defined in (6.1) , there exists some η = η ( ε ) > such that B ( X ) ≪ X log η X . To prove this result, we will need the following variant of a result of Ter¨av¨ainen [26] on the mean squareof the Dirichlet polynomial F ( s, χ ) := X X
Let ε > be sufficiently small. Define T = X / and F ( s, χ ) to be the Dirichletpolynomial defined in (6.5) , with P and δ > as in Section 3. Then, for T ≥ T , there exists some η = η ( ε ) > such that B ( X ) := X q ≤ Q qϕ ( q ) X χ ( q ) Z TT | F (1 + it, χ ) | dt ≪ Q log η X X q ≤ Q (cid:18) qT P log XX + qϕ ( q ) (cid:19) . (6.6) Proof of Proposition 6.4. First we consider when χ = χ as we have a different summand in this case. Wehave 2 qQ X x We first consider the contribution of the second and fourth terms, namely X P
XT qQ Z TT | F (1 + it, χ ) | dt ! , with T = X / and F ( s, χ ) := X X XT qQ Z TT | F (1 + it, χ ) | dt ! . We now apply Proposition 6.5. Note that we have P log X = Q , so that the first term in our bound for B ( X ) is bounded by ≪ XQ log η X X q ≤ Q (cid:18) P log XQ + qϕ ( q ) (cid:19) ≪ X log η X , as needed. For the second term, we want to bound X log XQ X q ≤ Q ϕ ( q ) max T ≥ XqQ T X χ ( q ) Z TT | F (1 + it, χ ) | dt. Applying Proposition 6.5, we have the bound ≪ X Q Q log η X X q ≤ Q max T ≥ XqQ (cid:18) P log XX + 1 T ϕ ( q ) (cid:19) ≪ X log η X , again using that P log X = Q . Overall we have that B ( X ) ≪ X log η X , for some η = η ( ε ) > 0, as required. (cid:3) Preliminaries on Dirichlet Polynomials Before we can prove Proposition 6.5, we first need the following preliminary lemmas on Dirichlet polyno-mials.7.1. Pointwise Bound. After we factorise our Dirichlet polynomial, there will be instances where the bestwe can do is use a pointwise bound. Before we state this bound, we need the following definition of awell-spaced set. Definition 7.1 (Well-Spaced Set) . We say a set T is well-spaced if for any t, u ∈ T with t = u we have that | t − u | ≥ Lemma 7.2 (Pointwise Bound) . Let S be a set of pairs ( t, χ ) with t ∈ [ − T, T ] and χ a Dirichlet character mod q which is well-spaced (i.e. if ( t, χ ) , ( u, χ ) ∈ S then | t − u | ≥ ). Suppose that min {| t | : ( t, χ ) ∈ S} ≫ log A N for all A > if χ = χ . Let P ( s, χ ) := X N we have | P (1 + it, χ ) | ≪ C N . Proof. This is [11, Lemma 10.7]. (cid:3) Definition 7.3 (Prime-factored polynomial, [26]) . Let M ≥ M ( s, χ ) = X M As in the work of Ter¨av¨ainen [26] and Matom¨aki, Radziwi l l[18], we take advantage of the bilinear structure to factorise our Dirichlet polynomial. Lemma 7.4 (Factorisation of Dirichlet Polynomials) . Define F ( s ) := X X This is [26, Lemma 2] (see also [18, Lemma 12]). (cid:3) ORRELATIONS OF ALMOST PRIMES 19 In some cases we will use the Heath-Brown identity to decompose a long polynomial into products ofshorter polynomials. Lemma 7.5 (Heath-Brown decomposition) . Let k ≥ be an integer, T ≥ and fix ε > . Define theDirichlet polynomial P ( s, χ ) := P P ≤ p such that L ≤ log C X and | P (1 + it, χ ) | ≪ (log C X )( | Q (1 + it, χ ) | + · · · + | Q L (1 + it, χ ) | ) for all t ∈ [ − T, T ] . Here, each Q j ( s, χ ) is of the form Q j ( s, χ ) = Y i ≤ J j M i ( s, χ ) , J j ≤ k, where each M i ( s, χ ) is a prime-factored Dirichlet polynomial (depending on j ) of the form X M i This is the Dirichlet character analogue of [26, Lemma 10], which follows the same argument. (cid:3) Mean Value Theorems for Dirichlet Polynomials. Now we state two mean value theorems, thefirst being the classical result: Lemma 7.6 (Mean Value Theorem) . Let q, X ≥ and let a n be arbitrary complex numbers with F ( s, χ ) := P X See, for example, [23, Chapter 6]. (cid:3) Next we state a variant of the mean value theorem which will allow us to save a log X in certain parts ofthe proof. Lemma 7.7. With the same assumptions as Lemma 7.6, we have that X χ ( q ) Z T − T | F ( it, χ ) | dt ≪ T ϕ ( q ) X X This is the Dirichlet character analogue of [26, Lemma 4], which follows from [14, Lemma 7.1]. Theproof is contained in the proof of [20, Lemma 5.2]. (cid:3) After factorising the Dirichlet polynomial F and splitting the domain of integration according to the sizeof the factors, there will be cases where the mean value is taken over a well-spaced set. In this case, we willapply the Hal´asz-Montgomery inequality: Lemma 7.8 (Hal´asz-Montgomery Inequality) . Let T ≥ , q ≥ . Let S be a well-spaced set of pairs ( t, χ ) with t ∈ [ − T, T ] and χ a Dirichlet character mod q . With the same assumptions as Lemma 7.6, we have that X ( t,χ ) ∈S | F ( it, χ ) | ≪ (cid:18) ϕ ( q ) q X + |S| ( qT ) / (cid:19) (log(2 qT )) X X This is [16, Lemma 7.4]. (cid:3) Large Value Theorems. There will be subsets of the domain of integration where a short Dirichletpolynomial factor is large, in which case we apply the following large value theorem. Lemma 7.9 (Large Value Theorem) . Let P ≥ , V > and F ( s, χ ) = P P
This is the Dirichlet character analogue of [26, Lemma 6] and [18, Lemma 8]. Also see [16, Lemma7.5]. (cid:3) Remark 7.10. As remarked in [26, Remark 6], this lemma can still be applied to polynomials with coeffi-cients not only supported on the primes as long as we have P ≫ X ε , as will be the case in our application.Alternatively, in the case that we have a longer Dirichlet polynomial factor which is large, we will applya result of Jutila on large values. Lemma 7.11 (Jutila’s Large Value Theorem) . Let F ( s, χ ) = P X This is the first bound of the main theorem in [15]. (cid:3) Moments of Dirichlet Polynomials. After decomposing the Dirichlet polynomial using Heath-Brown’s decomposition (Lemma 7.5), we can have a long polynomial which is the partial sum of a Dirichlet L -function (or its derivative). In this case, we will apply the Cauchy-Schwarz inequality to enable us to usethe following bound on the twisted fourth moment of such sums: Lemma 7.12 (Twisted Fourth Moment Estimate) . Let Q ≤ T ε , T ε ≤ T ≤ T , ≤ M, N ≤ T o (1) anddefine the Dirichlet polynomials N ( s, χ ) = X N Proof. This is the Dirichlet character analogue of [26, Lemma 9] and we follow the same argument. In thecase 1 ≤ qt ≤ N , we use partial summation and the hybrid result of Fujii, Gallagher and Montgomery [6] X n ≤ N χ ( n ) n it = δ χ ϕ ( q ) N it q (1 + it ) + O (( qτ ) / log( qτ )) , with τ := | t | + 2 in place of the zeta sum bound to get that X q ≤ Q ϕ ( q ) X χ ( q ) Z TT | N (1 + it, χ ) | | M (1 + it, χ ) | dt ≪ T o (1) max M We are now able to prove Proposition 6.5, completing the proof of Proposition 6.1 and consequentlyTheorem 1.2. We will adapt the argument appearing in [26, Sections 2-4]. We first factorise the Dirichletpolynomial F ( s, χ ) before bounding the contribution of the remainder terms, that is, the second term of(7.1). Lemma 8.1. Let ε > be sufficiently small and T = X / . Denote G v ( s, χ ) := X e vU , where we take I = [ U log P, (1 + δ ) U log P ] , U := Q ε and v ∈ I a suitable integer.Proof. We factorise F ( s, χ ) using Lemma 7.4 to get that F ( s, χ ) = X v ∈ I ∩ Z G v ( s, χ ) H v ( s, χ ) + X k ∈ [ Xe − /U ,Xe /U ]or k ∈ [2 X, Xe /U ] d k χ ( k ) k s , where I = [ U log P, (1 + δ ) U log P ], U := Q ε and | d k | ≤ X k = p p P
Returning to (8.2), by Mertens’ theorem we have that X Xe − /U ≤ p p ≤ Xe /U P
We will use Brun’s sieve to bound the second of these sums. We may trivially bound |{ n ≤ X : n = p p , p ∈ ( P, P δ ] }| ≪ |{ n ≤ X : n = p m, p ∈ ( P, P δ ] , ( m, P ( z )) = 1 }| where we define P ( z ) = Q p Brun’s sieve then gives the bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) m ∈ [ Xe − /U , Xe /U ] : (cid:18) m ( m + h ) , P ′ ( z )Π (cid:19) = 1 (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≪ X ( e /U − e − /U ) hϕ ( h ) Y p Combining these estimates and applying the definition of U , these two sums contribute ≪ (cid:18) ϕ ( q ) T log XX + 1 (cid:19) Q log η X to (8.1) for some η = η ( ε ) > 0, as needed. (cid:3) It remains to estimate the integral appearing in Lemma 8.1. We split the domain of integration [ T , T ]according to the size of the polynomial G v . We will first bound the contribution of S ⊂ [ T , T ] × { χ mod q } defined by S := { ( t, χ ) ∈ [ T , T ] × { χ mod q } : | G v (1 + it, χ ) | ≤ e − α v U } , (8.3)where α is defined according to the size of H as follows α := ( − ε ′ , if log ε X ≤ H ≤ exp((log X ) o (1) ) , − ε ′ , if exp((log X ) o (1) ) < H ≤ X log − A X, (8.4)where ε ′ > ε > 0. We may write S = [ χ mod q { χ } × T ,χ (8.5)for some T ,χ ⊂ [ T , T ].8.1. The contribution of S . We first treat the contribution of the integral over T ,χ , where the polynomial G v (1 + it, χ ) is pointwise small. Lemma 8.2. Let ε > be sufficiently small and T ,χ be defined as in (8.5) . Then, there exists some η = η ( ε ) > such that U log P X q ≤ Q qϕ ( q ) X χ ( q ) Z T ,χ | G v (1 + it ) | | H v (1 + it ) | dt ≪ Q log η X X q ≤ Q (cid:18) qP T log XX + 1 (cid:19) . Proof. First we apply the definition of T ,χ , bounding pointwise | G v (1 + it, χ ) | ≤ e − α v U ≤ P − α to boundthe integral over T ,χ by Z T ,χ | G v (1 + it, χ ) | | H v (1 + it, χ ) | dt ≪ P − α Z T ,χ | H v (1 + it, χ ) | dt. Applying Lemma 7.7, we have that X χ ( q ) Z T ,χ | G v (1 + it, χ ) | | H v (1 + it, χ ) | dt ≪ P − α T ϕ ( q ) e v /U X X Xev /U
For the second term, we have by Lemma 2.1 that X ≤ h ≤ XTev /U q | h X Xev /U
Combining these estimates, we have that X χ ( q ) Z T ,χ | G v (1 + it, χ ) | | H v (1 + it, χ ) | dt ≪ ϕ ( q ) P δ − α q log X (cid:18) qP T log XX + 1 (cid:19) . Thus the overall contribution to the sum B ( X ) is U P δ − α log P log X X q ≤ Q (cid:18) qP T log XX + 1 (cid:19) . Now, by our choices of P , U and the definition of α with ε ′ sufficiently small in terms of ε , we have that U P δ − α log P log X ≪ (log log X ) log (2+2 ε ) A ′ +( δ − α ) B − X ≪ Q log η X , for some η = η ( ε ) > δ > (cid:3) The contribution of the complement of S . It remains to consider the contribution of the com-plement of S . We apply Lemma 7.5 to H v (1 + it, χ ) with k = 3, decomposing this polynomial into | H v (1 + it, χ ) | ≪ (log C X ) ( | Q (1 + it, χ ) | + · · · + | Q L (1 + it, χ ) | ) , where L ≤ log C X for some C > 0. Each Q j ( s, χ ) is of the form Q j ( s, χ ) = Q i ≤ J j M i ( s, χ ) with J j ≤ ≤ j ≤ L , where M i ( s, χ ) are prime-factored Dirichlet polynomials of the form X M i Suppose we have Q j ( s, χ ) = Q i ≤ J j M i ( s, χ ) for some 1 ≤ j ≤ L with exp (cid:16) log X log log X (cid:17) ≪ M i ≤ X / o (1) for each i ≤ J j ≤ 6. Then, we rewrite Q j ( s, χ ) = M ( s, χ ) M ( s, χ ) with exp (cid:16) log X log log X (cid:17) ≪ M ≤ X / o (1) and M = X o (1) /M . Note we may write M = X ν + o (1) for some 0 < ν ≤ / o (1). Ifthe polynomial X M i Otherwise, we can have that at most two of the lengths M i satisfy M i > X / o (1) , inwhich case we may write Q j ( s, χ ) = N ( s, χ ) N ( s, χ ), having trivially bounded the remaining polynomials.Here, each N i ( s, χ ) is of the form X N i To treat the contribution of these sums, we split the complement of S according tothe size of M (1 + it, χ ): S := { ( t, χ ) ∈ [ T , T ] × { χ mod q } : | M (1 + it, χ ) | ≤ M − α } \ S , S := ([ T , T ] × { χ mod q } ) \ ( S ∪ S ) , (8.6)with α := − ε ′ > α . As before, we may write S = [ χ mod q { χ } × T ,χ , S = [ χ mod q { χ } × T χ , (8.7)for some T ,χ , T χ ⊂ [ T , T ]. We first consider the contribution of the integral over T ,χ . Lemma 8.3. Let ε > be sufficiently small. Let T ,χ be defined as in (8.7) and M ( s, χ ) , M ( s, χ ) be theprime-factored polynomials defined previously. Then X q ≤ Q qϕ ( q ) X χ ( q ) Z T ,χ | G v (1 + it, χ ) | | M (1 + it, χ ) M (1 + it, χ ) | dt ≪ X − o (1) . Proof. By definition of T ,χ , we have that ( | G v (1 + it, χ ) | P α ) k − ≥ 1, where k = ⌈ log M / log P ⌉ .Therefore, we have X χ ( q ) Z T ,χ | G v (1 + it, χ ) | | M (1 + it, χ ) M (1 + it, χ ) | dt ≪ M − α P α ( k − X χ ( q ) Z T ,χ | G kv (1 + it, χ ) M (1 + it, χ ) | dt. (8.8)By the choice of k , we have that P α k ≪ exp (cid:18) α log P log M log P (cid:19) = M α . Therefore (8.8) is bounded by ≪ M α − α P − α X χ ( q ) Z T ,χ | G kv (1 + it, χ ) M (1 + it, χ ) | dt ≪ M α − α P − α X χ ( q ) Z T ,χ | A (1 + it, χ ) | dt, where we define A ( s, χ ) := X n ∈ J A n χ ( n ) n s , with J := ( M e kv /U , M e k ( v +1) /U ] and the coefficients A n satisfying | A n | ≤ X n = p ··· p k me v /U
5, as before. Note that the primes p , . . . , p k are not necessarily distinct and m may also haveprime factors in the range ( e v /U , e ( v +1) /U ]. Applying Lemma 7.6 to the integral, we have that X χ ( q ) Z T ,χ | A (1 + it, χ ) | dt ≪ (cid:18) ϕ ( q ) T + ϕ ( q ) q M e kv /U (2 e k/U − (cid:19) X n ∈ J ( n,q )=1 | A n | n . Following [18, Lemma 13], for the coefficients A n of A ( s, χ ), we have the bound | A n | ≪ M o (1)2 X n = p ··· p k me v /U
Since v ∈ I , we have that e kv /U ≫ P k ≫ M by the definition of k . We also have that 2 e k/U − ≪ X χ ( q ) Z T ,χ | A (1 + it, χ ) | dt ≪ (cid:18) ϕ ( q ) TM M + ϕ ( q ) q (cid:19) X o (1) M /B ≪ ϕ ( q ) X o (1) M /B , as we have M M = X o (1) and T ≤ X o (1) . Returning to (8.8), we have the bound X χ ( q ) Z T ,χ | G v (1 + it, χ ) | | M (1 + it, χ ) M (1 + it, χ ) | dt ≪ ϕ ( q ) M /B +2 α − α P − α X o (1) . With our choices of α , α and B , we have that 12( α − α ) < B. Summing over q introduces a factor of Q . Writing M = X ν + o (1) , for some 0 < ν ≤ / o (1), we find that X q ≤ Q qϕ ( q ) X χ ( q ) Z T ,χ | G v (1 + it, χ ) | | M (1 + it, χ ) M (1 + it, χ ) | dt ≪ Q P − α X ( ν + o (1))(1 /B +2 α − α )+ o (1) , and choosing ε ′ > ε > α , α ensures the above isbounded by X − o (1) , as needed. (cid:3) We now treat the contribution of the integral over T χ , applying the Hal´asz-Montgomery inequality andthe large value theorems. Lemma 8.4. Let T χ be defined as in (8.7) . Let E > be sufficiently large. Then, we have that X q ≤ Q qϕ ( q ) X χ ( q ) Z T χ | G v (1 + it, χ ) M (1 + it, χ ) M (1 + it, χ ) | dt ≪ E X . Proof. Let M = X ν + o (1) for some 0 < ν ≤ / o (1). We first replace the integral over T χ with a sum overa well-spaced set. For each character χ mod q , cover T χ with intervals of unit length and from each intervaltake the point which maximises the integral over that interval. This set is not yet necessarily well-spaced,but we can split it into O (1) well-spaced subsets. Therefore we may write X χ ( q ) Z T χ | G v (1 + it, χ ) M (1 + it, χ ) M (1 + it, χ ) | dt ≪ X ( t,χ ) ∈T ′ | G v (1 + it, χ ) M (1 + it, χ ) M (1 + it, χ ) | , where T ′ is the well-spaced subset which maximises the right hand side. We now apply the prime-factoredproperty | M (1 + it, χ ) | ≪ log − F ′ X with F ′ > X ( t,χ ) ∈T ′ | G v (1 + it, χ ) M (1 + it, χ ) M (1 + it, χ ) | ≪ log − F ′ X X ( t,χ ) ∈T ′ | G v (1 + it, χ ) M (1 + it, χ ) | ≪ log − F ′ X (cid:18) ϕ ( q ) q X − ν + o (1) + |T ′ | ( qT ) / (cid:19) X e v /U r ≤ 5. If we can show that |T ′ | ≪ X / − ν − ε , then we will have that X χ ( q ) Z T χ | G v (1 + it, χ ) M (1 + it, χ ) M (1 + it, χ ) | dt ≪ log − E ′ X for some suitable E ′ > 0. Summing over q , we have that X q ≤ Q qϕ ( q ) X χ ( q ) Z T χ | G v (1 + it, χ ) M (1 + it, χ ) M (1 + it, χ ) | dt ≪ E X , where E > |T ′ | ≪ X / − ν − ε . Applying Lemma 7.11 with M (1 + it, χ ) l , V = M − α l , k = 2 , and l ∈ { , } , we have that |T ′ | ≪ X o (1) (cid:16) M α l + X M l (2 α − + XM l (4 α − (cid:17) ≪ ( X max( ν, − ν, − ν ) − ε , ( l = 2) ,X max( ν, − ν, − ν ) − ε , ( l = 3) . We have that ν ≤ + o (1). The inequality ν ≤ − ν holds when ν ≤ and we have that 2 − ν ≥ ν when ν ≤ . Note that 2 − ν ≤ − ν fails if ν < , so the inequality with l = 2 provides the bound |T ′ | ≪ X / − ν − ε in the range ≤ ν ≤ + o (1).Similarly, ν ≤ − ν holds for ν ≤ and 2 − ν ≥ ν when ν ≤ . We have that 2 − ν ≤ / − ν fails when ν < , so the inequality with l = 3 gives the required bound for |T ′ | when ≤ ν ≤ .For the remaining range ν < , we apply Lemma 7.9 with V = M − α to get that |T ′ | ≪ ( qT ) α X να + o (1) ≪ X (1+ ν )+100 ε ≪ X / − ν − ε , as required. (cid:3) Type I Sums. It remains to treat the contribution of the sums of form (7.3), applying the Cauchy-Schwarz inequality and a result on the twisted fourth moment of partial sums of Dirichlet L -functions. Lemma 8.5. Let ε > be sufficiently small. With N ( s, χ ) , N ( s, χ ) as defined above, we have that X q ≤ Q qϕ ( q ) X χ ( q ) Z [ T ,T ] \T ,χ | G v (1 + it, χ ) | | N (1 + it, χ ) N (1 + it, χ ) | dt ≪ X − ε/ . Proof. We split the domain of integration into dyadic intervals [ T , T ] such that T ≤ T ≤ T . As we arein the complement of S , we have that | G v (1 + it, χ ) P α | l − ≥ 1, where we define l = ⌊ ε log X/ log P ⌋ .Therefore, we have that X q ≤ Q qϕ ( q ) X χ ( q ) Z ([ T ,T ] \T ,χ ) ∩ [ T , T ] | G v (1 + it, χ ) | | N (1 + it, χ ) N (1 + it, χ ) | dt ≪ P α ( l − X q ≤ Q qϕ ( q ) X χ ( q ) Z T T | G v (1 + it, χ ) | l | N (1 + it, χ ) N (1 + it, χ ) | dt. Applying the Cauchy-Schwarz inequality three times (to the integral and the sums over χ and q ), we havethat the sum over q above is bounded by ≪ X q ≤ Q ϕ ( q ) X χ ( q ) Z T T | G v (1 + it, χ ) | l | N (1 + it, χ ) | dt X q ≤ Q q ϕ ( q ) X χ ( q ) Z T T | N (1 + it, χ ) | dt . (8.9) ORRELATIONS OF ALMOST PRIMES 29 We apply Lemma 7.6 to the second integral. Noting that N (1 + it, χ ) either has coefficients 1 or log n , wefind that X q ≤ Q q ϕ ( q ) X χ ( q ) Z T T | N (1 + it, χ ) | dt ≪ X o (1) X q ≤ Q q (cid:18) T + N /qN (cid:19) ≪ X o (1) (cid:18) T + N N (cid:19) . We next treat the first term appearing in (8.9). We have that | G v (1 + it, χ ) | l = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X e v /U
1. Returning to (8.9), we have that X q ≤ Q qϕ ( q ) X χ ( q ) Z ( T \T ) ∩ [ T , T ] | G v (1 + it, χ ) | | N (1 + it, χ ) N (1 + it, χ ) | dt ≪ P α ( l − X o (1) ( l !) o (1) (cid:18) Q T N P l + 1 T (cid:19) / (cid:18) T + N N (cid:19) / ≪ P α ( l − X o (1) ( l !) o (1) (cid:18) Q T N N P l ( T + N ) + 1 N + 1 T (cid:19) / . We have that N N = X o (1) with N ≥ X / − o (1) and X / o (1) ≤ N ≤ X / o (1) . As we also have that X / = T ≤ T ≤ T ≤ X o (1) , the above is bounded by ≪ P α ( l − X o (1) ( l !) o (1) P l + 1 N + 1 T / ! . Summing the contribution of each of the integrals over the dyadic intervals multiplies the above estimate bylog X . By the definition of l , we have that( l !) o (1) ≪ (log X ) l (1+ o (1)) ≪ exp (cid:18) (2 + o (1)) ε log X log log XB log log X (cid:19) ≪ X (2+ o (1)) ε/B and we also have that( P α − log X ) l (1+ o (1)) = log l ( B (2 α − o (1)) X ≪ exp (cid:18) ε ( B (2 α − 1) + 2)(1 + o (1)) log log X log XB log log X (cid:19) ≪ X ε ( B (2 α − o (1)) B ≪ X − ε/ . Overall we have the bound X q ≤ Q qϕ ( q ) X χ ( q ) Z [ T ,T ] \T ,χ | G v (1 + it, χ ) | | N (1 + it, χ ) N (1 + it, χ ) | dt ≪ X − ε/ , as needed. (cid:3) Completing the proof of Proposition 6.5. We may now combine these estimates to complete theproof of Proposition 6.5. Proof of Proposition 6.5. By Lemma 8.1, we have that X q ≤ Q qϕ ( q ) X χ ( q ) Z TT | F (1 + it, χ ) | dt ≪ X q ≤ Q qU log Pϕ ( q ) X χ ( q ) Z TT | G v (1 + it, χ ) | | H v (1 + it, χ ) | dt + 1 Q log η X (cid:18) qT log XX + qϕ ( q ) (cid:19) ! ≪ X q ≤ Q qU log Pϕ ( q ) X χ ( q ) Z T ,χ | G v (1 + it, χ ) | | H v (1 + it, χ ) | dt + Z [ T ,T ] \T ,χ | G v (1 + it, χ ) | | H v (1 + it, χ ) | dt ! + 1 Q log η X (cid:18) qT log XX + qϕ ( q ) (cid:19) ! for some η = η ( ε ) > v ∈ I . We apply Lemma 8.2 to bound the contribution ofthe integral over T ,χ , finding that the above is bounded by ≪ X q ≤ Q qU log Pϕ ( q ) X χ ( q ) Z [ T ,T ] \T ,χ | G v (1 + it, χ ) | | H v (1 + it, χ ) | dt + 1 Q log η X (cid:18) qT P log XX + qϕ ( q ) (cid:19) ! . Combining Lemmas 8.3 to 8.5, we bound the contribution of the complement of S by ≪ U log P log E X ≪ F X for some sufficiently large F > 0, which is negligible. Thus, we have that X q ≤ Q qϕ ( q ) X χ ( q ) Z TT | F (1 + it, χ ) | dt ≪ Q log η X X q ≤ Q (cid:18) qT P log XX + qϕ ( q ) (cid:19) , as required. (cid:3) Proof of Theorem 1.5 We now briefly outline the adjustments we make to the proof of Theorem 1.2 in order to prove Theorem 1.5.We will once again apply the Hardy-Littlewood circle method and in (3.5) and (3.7) we take Q := log A ′ X, A ′ > , Q := P log C X, H ≥ Q log D X, (9.1)where C is chosen sufficiently large in terms of A ′ and D is chosen sufficiently large in terms of A ′ and C . InLemma 8.1 we instead define I := [ U log P , U log P ] where U := Q E , E > α := ε ′ > ε > p ∈ [ P , P ] into dyadic intervals before applying the inequality. We now outline how to modify theproof of Proposition 3.2. ORRELATIONS OF ALMOST PRIMES 31 Proposition 9.1. Let A > , B > and m be defined as in (3.7) with Q , Q as in (9.1) . Let Q log D X ≤ H ≤ X log − A X with D > sufficiently large. For α ∈ m we have that Z m ∩ [ α − H ,α + H ] | S ( θ ) | dθ ≪ X log B X . Proof. As before, we apply Lemma 2.2 to the minor arc integral so that we need to bound I ≪ H Z XX (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x 1, as claimed. (cid:3) Proposition 9.2. Let A > , B > . Let exp((log X ) − o (1) ) ≤ H ≤ X log − A X . Let M be defined as in (3.5) with Q , Q as in (9.1) . Then, for all but at most O ( HQ − / ) values of < | h | ≤ H we have that Z M | S ( α ) | e ( − hα ) dα = S ( h ) X X P ≤ p ≤ P p + O (cid:18) X log B X (cid:19) , where S ( h ) is the singular series given in Theorem 1.2.Proof. Recalling Lemma 5.1, we have the expansion S ( α ) = µ ( q ) ϕ ( q ) X P
The proof of Proposition 6.3 requires modifying in a similar way to Proposition 3.2. In (6.4) we split thesum over P ≤ p ≤ P into dyadic intervals P < p ≤ P before applying Cauchy-Schwarz, Lemma 2.3 andthen combining the contributions of the dyadic sums. (cid:3) We are now able to complete the proof of Theorem 1.5. Proof of Theorem 1.5. By partial summation and Mertens’ theorem we have the bound Z | S ( α ) | dα = X X Proof of Theorem 1.6 We outline the modifications needed to prove Theorem 1.6. When applying the Hardy-Littlewood circlemethod, in (3.5) and (3.7) we now choose Q := log A ′ X, A ′ > , Q := X / ε/ , H ≥ QX ε/ . (10.1)As in Section 9, in Lemma 8.1 we instead define I := [ U log P , U log P ] where U := Q E , E > α := ε ′ > ε > 0. Analogously to the almost prime case,we may write X X 1) we define the exponential sum S ′ ( α ) := P X Let A > , B > and let ε > be sufficiently small. Let X / ε ≤ H ≤ X log − A X . Let M be defined as in (3.5) with Q , Q as in (10.1) . Then, for all but at most O ( HQ − / ) values of < | h | ≤ H we have that Z M S ( α ) S ′ ( α ) e ( − hα ) dα = S ( h ) X X P ≤ p ≤ P p + O (cid:18) X log B X (cid:19) , where S ( h ) is the singular series given in Theorem 1.2.Proof. We can expand S ′ in terms of Dirichlet characters (see for example [22]): S ′ ( α ) = µ ( q ) ϕ ( q ) X X 0, as required. Combining this with our estimates for A ( X ) , B ( X ) (from Proposition 9.2) and C ( X )we have that the error term in (10.2) is O ( X log − B X ), as required. (cid:3) Proof of Theorem 1.6. Analogously to the proof of Theorem 1.2, by [19, Proposition 3.1] we have that X < | h |≤ H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X X Therefore, applying Chebyshev’s inequality and Proposition 10.1 gives the result. (cid:3) Acknowledgements The author is grateful to her supervisor Stephen Lester for suggesting the problem and for many helpfulcomments and discussions throughout this work. 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