Bounded gaps between product of two primes in imaginary quadratic number fields

We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of Goldston–Graham–Pintz–Yildirim (Proc Lond Math Soc 98:741–774, 2009), and Maynard (Ann Math 181:383–413, 2015). An important consequence of our main theorem is existence of infinitely many pairs α1,α2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1, \alpha _2$$\end{document} which are product of two primes in the imaginary quadratic field K such that |σ(α1-α2)|≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\sigma (\alpha _1-\alpha _2)|\le 2$$\end{document} for all embeddings σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} of K if the class number of K is one and |σ(α1-α2)|≤8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\sigma (\alpha _1-\alpha _2)|\le 8$$\end{document} for all embeddings σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} of K if the class number of K is two.


Introduction
One of the classical problems in prime number theory is to study the gaps between prime numbers. The famous twin prime conjecture asserts that there are infinitely many pairs (p 1 , p 2 ) of prime numbers such that |p 1 − p 2 | = 2. Although this conjecture remains out of reach, study of this conjecture leads to several interesting results. The first and very important breakthrough in this direction is the result of Goldston et al. [3] who showed that lim inf n→∞ p n+1 − p n log p n = 0 , where p n denotes the nth prime. Zhang [16] has subsequently improved this result by showing that lim inf n→∞ (p n+1 − p n ) ≤ 7 × 10 7 .
Very shortly afterwards, a further breakthrough was obtained by Maynard [9], who developed a multidimensional version of the Selberg sieve to obtain 600 instead of 7 × 10 7 . DHJ polymath [13] group extends the methods of Maynard by generalizing the Selberg sieve further reduce it to 246 unconditionally, and 6 under the assumption of the generalized Elliott-Halberstam conjecture.
Let {q n } n≥1 be the sequence of positive integers which are products of exactly two primes written in the increasing order. The members of this sequence are called E 2 numbers. Heuristically problems involving E 2 -numbers are as difficult as problems involving prime numbers as sieve methods do not seem to distinguish between numbers with even numbers of prime factors and odd number of prime factors (parity principle of Selberg [14]). Hence it is interesting to study the gaps between E 2 -numbers. This study was initiated by Goldston et al. [4] who showed that lim inf n (q n+1 − q n ) ≤ 26 . (1.1) Later developing the methods of [4], they are able to improve the constant on the right hand of (1.1) to 6 in [5]. Let K be a number field and let O K be its ring of integers. We say that an element α ∈ O K is prime if the principle ideal αO K is a prime ideal. Castillo et al. [1] have initiated the study of gaps between primes in number fields. By extending the methods of Maynard-Tao they showed that for a totally real field K there are infinitely many primes α 1 and α 2 in O K such that |σ (α 1 − α 2 )| ≤ 600 for every embedding σ of K . The case when K is imaginary is first considered by Vatwani [15]. In particular it is shown in [15] that there are infinitely many prime pairs (p 1 , p 2 ) ∈ Z[i] × Z[i] such that N (p 1 − p 2 ) < 246 2 , where N (·) denotes the norm on Q(i). The method of the proof can be generalized to cover any imaginary quadratic number field with class number 1.
In the spirit of [1,4,5,8] it is natural to consider gaps between products of two primes in number fields. Before stating our main result of this article, we will fix some notations. Let P be the set of prime numbers in O K . Let G K 2 be the set of all α ∈ O K which can be written as a product of two elements from P. We say that a tuple (h 1 , . . . , h k ) ∈ O k K is admissible if it does not cover all the residue classes modulo p for any prime ideal p of O K . Now we are in a position to state the main result of this paper. Theorem 1 Let K be an imaginary quadratic number field and let r ≥ 2 be an integer. Then there exists a positive integerk :=k(r, K ) such that for any admissible k-tuple (h 1 , . . . , h k ) ∈ O k K with k ≥k, there are infinitely many α ∈ O K such that at least r of α + h 1 , . . . , α + h k are G K 2 -numbers.
It is clear from Theorem 1.1 that lim inf |σ (α − β)| ≤ M(K ) where M(K ) is a constant depends only on K and the lim inf is taken when α, β runs over all G K 2 numbers. It will be clear at the end of the proof that the constant depends only on the class number. In the following corollaries we will precisely give the value of M(K ) when the class number is 1 or 2. There exist infinitely many G This article is organized as follows. In Sect. 2 we provide the necessary preliminaries to prove Theorem 1. In Sect. 3 we prove a variant of Bombieri-Vinogradov theorem for G K 2 -numbers. In Sect. 4 we explain the method of the proof. Section 2 is devoted to prove Proposition 2. In Sect. 5 we will prove some preparatory lemmas which are essential for the proof. In Sect. 6 we will choose the appropriate weights. In Sect. 8 we will conclude the proofs of Theorem 1, Corollary 1 and Corollary 2.

Notations and preliminaries
Here and in what follows, K denotes an imaginary quadratic field unless otherwise mentioned. For much of this article, we follow the notations of Hinz [6] and Castillo et al. [1]. Being an imaginary quadratic field K has no real embeddings and it has exactly two complex embeddings, namely σ 0 (the identity) and σ (complex conjugation). We observe that for any non-zero α ∈ O K , |σ (α)| ≥ 1. For N > 1, let Further, for N 1 < N 2 , we define We would also use A(N ) and P(N ) for A(2N, N ) and P (2N, N ) respectively. For a set S, |S| denotes its cardinality, for an element α ∈ K and an ideal q of O K , |α| and |q| denote the respective norms.
Remark 2 A clarification about the notations is much called for at this point. For an element α ∈ O K , |α| denotes its norm whereas |σ (α)| denotes absolute value as a complex number. For imaginary quadratic fields, they are related by Hence A 0 (N ) as defined above can also be described as These usages will be clear from the context as we proceed.
For elements a, b ∈ O K and an ideal q of O K , we write a ≡ b mod q to mean that the ideal generated by a − b is contained in q, i.e (a − b) ⊂ q. Moreover, if the ideal (a) generated by a ∈ O K does not have any common factor with q then we write (a, q) = 1. Given a non-zero ideal q ⊆ O K , we define analogues of three classical multiplicative functions, namely the norm |q| := |O K /q|, the Euler phi-function ϕ(q) := |(O K /q) × | and the Möbius function μ(q) := (−1) r if q = p 1 . . . p r for distinct prime ideals p 1 , . . . , p r and μ(q) = 0 otherwise. We use τ k (q) to denote the number of ways of writing q as a product of k factors and ω(q) to denote the number of distinct prime ideals containing q. For ideals a, b, we use [a, b] and (a, b) to denote LCM and GCD of a, b. The k-tuple (a 1 , . . . , a k ) with a j ∈ O K for all j (1 ≤ j ≤ k) is denoted by a. We use w 1 , w 2 to denote prime elements of O K . For any R ∈ R, |a| ≤ R is to be interpreted as k j=1 |a j | ≤ R. The notion of divisibility among k-tuples is defined componentwise, i.e, For any integral ideal q of O K , a|q ⇔ k j=1 a j |q. We use the notation [a, b] to denote the product of the component-wise least common multiples, i.e. [a, b] = k j=1 [a j , b j ] and (a, b) = 1 to mean that the ideals a and b are coprime, where 1 is the trivial ideal. For Re(s) > 1, the Dedekind zeta function of K is defined by where the sum is over all non-zero ideals of O K . This function admits meromorphic continuation to the whole complex plane with a pole at s = 1. Let c K denote its residue at s = 1. Now we note that [1, page 4] the number of elements α ∈ A(N ) satisfying a congruence condition α ≡ α 0 (mod q) is given by The following lemma is central in estimation of the sums that arise in Selberg's higher dimensional sieve.
where c K := Res s=1 ζ K (s) and the singular series The following lemma is a consequence of Minkowski's lattice point theorem (see [1, page 12]).

Lemma 2 Let
A 0 (N ) and A(N ) be defined as above. We have Let ω K be the number of roots of unity contained in K and h K be the class number of K . The following lemma is a special case of Mitsui's generalized Prime number theorem [10].

Lemma 3
Let P 0 (N ) be defined as above. We have where c is a non-zero positive real number.
We denote m K := ω K h K R K as Mitsui's constant. As a direct consequence of Lemma 3 we get Lemma 4 Let P 0 (N ) be defined as above. Then we have We shall also use Dedekind's class number formula.

Lemma 5 ([12]
, Corollary 5.11 ) Let c K , ω K and h K be defined as above. We have

Lemma 6
Let K be an algebraic number field. For any natural number R ≥ 2, we have where first sum is over all non-zero integral ideals of O K whose norm is less than or equal to R.

A generalization of the Bombieri-Vinogradov theorem
Most important case is when S = P. In this case, an analog of Elliott-Halberstam conjecture for number fields predicts that the inequality (3.1) holds with any ϑ in 0 < ϑ ≤ 1.
Hinz [6] showed that primes have level of distribution 1 2 in totally real algebraic number fields. Huxley [7] obtained level of distribution 1 2 for a weighted version of (3.1). The G K 2 -numbers for K = Q was shown by Motohashi to have level of distribution 1 2 . For our purposes, it is convenient to define the following related quantities.
Using a theorem of [7] and following the argument in Lemma 10.2 of [15], we prove the following generalization of the Bombieri-Vinodradov theorem.

Proposition 1 Let K be an imaginary quadratic number field. Then (3.1) holds for any
Proof Let q be an ideal in O K . We denote the ray class group (mod q) by C q and a ray class by L q . Let π(x, K ) be the number of prime ideals in O K of norm ≤ x and χ P be the characteristic function of the prime ideals in O K . We define where h(q) denotes the cardinality of the ray class group C q . We will now use the following lemma. [7]) Using the notations as above, for any A > 0, there exists a real number B > 0 such that for any ϑ ≤ 1 2 we have

Lemma 7 (Huxley
We have the following relation between the number of ray classes and the class number ( [15]): where U is the unit group of O K , U q,1 = {α ∈ U : α ≡ 1 (mod q) , α 0} and h K is the class number of K where α 0 means all the real conjugates (if any) of α are positive. Now we will estimate the index set [U : U q,1 ]. To do that we define the following homomorphism . Then the kernel of ψ is U q,1 and image of ψ is the residue classes (mod q) that contain a unit. Let T q = Im(ψ). Then |T q | = [U : U q,1 ] and h(q) ϕ(q) = h |T q | . Since number of units in a imaginary quadratic number field is 2, 4 or 6, so if u 1 , u 2 ∈ U satisfies u 1 ≡ u 2 (mod q) then |q| must divide |u 1 − u 2 |, which is atmost 4. Thus for |q| > 4 we see that T q = |U |, which only depends only on K and not on q. Therefore using these estimates, from Lemma 7 we obtain the following. Lemma 8 Using the notation as in Lemma 7, for any A > 0 there exists a positive real number B such that for any 0 < ϑ ≤ 1 2 , we have Proof Let a ∈ O K , (a, q) = 1 and L q (a) be the ray class containing (a). Then from (3.2) we get It is easy to see that all integral ideals belonging to L q (a) are principal. Therefore we obtain We also observe that there is an one to many correspondence between {η ∈ P, |η| ≤ x, (η) ∈ L q (a)} and {w ∈ P, |w| ≤ x, w ≡ a(q)} depending on the number of units in O K (see [ [15], Sect. 10] for more details). More precisely, we have for any ϑ ≤ 1 2 and for any A > 0. Now Prime ideal theorem tells us Also from Lemma 4 and using ω K = |U |, we get Combining (3.5) and (3.6) we obtain Also note that From (3.7) and (3.4) we complete proof of the proposition.
We would use the above result in the following form which can be easily deduced by partial summation.

Lemma 9 Let K be an imaginary quadratic number field. For any
For the function β, we define An arithmetic function f is said to have level of distribution ϑ for 0 < ϑ ≤ 1 if for any Let τ (n) be the number of divisors of a natural number n. A complex valued arithmetic function f is said to satisfy Siegel-Walfisz condition if there exist positive constant C such that holds for all D > 0 and for any non-principal Dirichlet character χ (mod q) with q (log x) D .
If arithmetic functions f and g both satisfy (3.9) and have level of distribution 1 2 then Motohashi [11] obtained that the Dirichlet convolution f * g also does so. In [2], we extend Motohashi's [11] result to arithmetic functions on imaginary quadratic number fields. As the proof can be carried forward for any level of distribution 0 < ϑ ≤ 1 2 , viewing β as a Dirichlet convolution of characteristic functions of P(Y , N b ) and P(N b , ∞), we get the following lemma. More precisely, it is a direct application of Cauchy-Schwarz inequality and Corollary 1.5 of [2].

Lemma 10 Let K be an imaginary quadratic number field. For
(3.10)

Method
Now we will describe the method of proof which is a combination of methods of [5] and [9].
The main objects of consideration are the sums where the inner sum is a k-fold sum over integral ideals and λ d 1 ,...,d k are suitably chosen weights to be made explicit later.
Since each summand is non-negative, if we can show that S 2 > ρS 1 for some positive ρ, then there must be at least one α ∈ A(N ) such that among α + h 1 , . . . , α + h k atleast [ρ] + 1 are G K 2 -numbers. We choose the weights λ d 1 ,...,d k in such a way that λ d 1 ,...,d k = 0 unless (d i , m) = 1, d i is square-free, and |d 1 · · · d k | ≤ R for each i = 1, · · · , k, where R will be chosen later to be a small power of N . The main result of this section is the following. Proposition 2 Let K be an imaginary quadratic number field. Suppose that the primes P and G K 2 -numbers have a common level of distribution 0 < ϑ ≤ 1, and set R = N ϑ (log N ) −C for some constant C > 0. For a given a piecewise differentiable function F : . − x k and T m (y) = min(y, T m ).

Preparations
The sum S 1 has been calculated in [1, Proposition 2.1]. So we would only work with S 2 . By squaring innermost sum and interchanging summation from Eq. (4.1) we can write S 2 as We note that [a i , b i ] and [a j , b j ] are relatively coprime for i = j since the primes dividing Hence we conclude that either [a m , b m ] = 1 or [a m , b m ] = (w 1 ). Before discussing either of these cases we need the following lemma.

Lemma 11
For any function f : Since |f | ≤ 1, we get

The case [a m , b m ] = (w 1 )
In this case w 1 ∈ P(Y , R ) with R = R 1/2 because of the support of λ a and λ b . Let w 1 be the inverse of w 1 (mod q/(w 1 )) . Similarly as above Now α ∈ A(N ) and α = w 1 w 2 . So we separate the above sum with respect to primes w 1 and w 2 . We note that w 1 w 2 ∈ A(N ) if and only if w 2 ∈ A(N /|w 1 | 1/2 ). Therefore in this case, we have For each q, the number of ways of choosing a 1 , . . . , a k and b 1 , is at most τ 3k (q). Therefore for each 1 ≤ m ≤ k, from Eq. (5.1), the sum S 2m can be written as where λ max = sup a |λ a |. Using Lemma 6, it can seen that the first error term of the above expression of S 2m is bounded above by Lemma 10 gives that the second error term of S 2m is bounded above by λ 2 max |A(N )| (log N ) B for any B > 0.
Lemma 9 gives that the third error term of S 2m is bounded above by Combining these estimations of error terms we get the following lemma.

Lemma 12 Let S 2m be defined as in (5.1). Then with the hypothesis of Proposition 2 we have
We define (5. 2) The sum S 2m (w 1 ) is estimated in the following lemma.

Lemma 13
Let S 2m (w 1 ) be defined as in (13). For ideals r 1 , . . . , r k of O K , we define y (m) where g is the multiplicative function defined by g(p) = |p| − 2 for all prime ideals p of A. Let y (m) max (w 1 ) = sup r 1 ,...,r k |y (m) r 1 ,...,r k (w 1 )|. Then we have Proof From the definition of g it follows that . λ a 1 ,...,a k is supported on ideals a 1 , . . . , a k with (a i , m) = 1 for each i and (a i , a j ) = 1 ∀i = j. Thus we may drop the requirement that m is coprime to each of the [a i , b i ] from the summation, since these terms have no contribution. Thus the only remaining restriction is that (a i , b j ) = 1 ∀i = j. So we can remove this coprimality condition by Möbius inversion to get

Note that
. Now we make the following change of variables: From Lemma 6 the above quantity is bounded above by The main term of S 2m (w 1 ) is obtained from s i,j = 1 for all i = j which completes the proof.
For ideals r 1 , . . . , r k , we define Therefore λ max y max (log R) k . The following lemma gives a relation between the quantities y (m) r 1 ,...,r k (w 1 ) and y r 1 ,...,r k .
Proof Using (5.4), we get where 1 (w 1 )|r m is the indicator function which takes value 1, if (w 1 ) | r m and 0 otherwise. We see from the support of y r 1 ,...,r k that we may restrict the summation over r j to (r j , m) = 1. The main term is given by r j = u j ∀j, for all other terms there exists j = m such that |r j | > D 0 |u j |. Therefore the error term is bounded above by The main term given by r j = u j ∀j = m is

Choosing the weights
For a real valued piecewise differentiable function F on R k as in Proposition 2 we define Note that, y r 1 ,...,r k is supported on square-free r = k i=1 r i such that (r, m) = 1. Hence Estimation of S 1 . To use Lemma 1 we set .
The singular series in Lemma 1 is easily computed to be and also L log log R. Thus we get where c K = Res s=1 ζ K (s). Estimation of S 2 . Observe that Therefore by Lemma 1, we get Putting S 1 and S 2 together, we get u 1 ,...,u k (w 1 ) (6.1)

Proof of Proposition 2
Using the value of y (m) u 1 ,...,u k (w 1 ) given by Eq. (6.1) in Lemma 13 we get Setting Y := Y 1/2 and the above equation in Lemma 12, we have Using Lemma 4 we can say that π Using this the main term of S 2m becomes Estimation of S 4 and S 5 . To calculate both S 4 and S 5 we use Lemma 1 with The singular series can be easily computed to be S = ϕ(m) |m| + O ϕ(m) |m|D 0 and also L log D 0 . Recalling the coprimality conditions S 4 can be written as We note that, two ideals a and b with (a, m) = (b, m) = 1 but (a, b) = 1 must have a common prime factor with norm greater than D 0 . Thus we can drop the requirement that u i , u j = 1, at the cost of an error of size Thus it is enough to evaluate the following sums Using Lemma 1 we have, Finally it remains to calculate the following sums From the above estimations, we get Putting u = R y , first integral S 8 gives main term for S 7 which is Second integral S 9 giving error term of S 7 can be estimated as where B = 2 ϑ as defined in Proposition 2. Therefore combining these estimations we have By using the same method we have Therefore we conclude that Recall that Therefore Proposition 2 follows from 7.1.

proof of theorem 1, corollary 1 and corollary 2
We start with the following corollary of the Proposition 2.

Corollary 3 Let K be an imaginary quadratic number field and m K be its Mitsui constant.
Suppose that the primes P and G K 2 -numbers have a common level of distribution 0 < ϑ ≤ 1.  Then there are infinitely many α ∈ O K such that at leastr k of the α + h 1 , . . . , α + h k are G K 2 -numbers.
Proof Since each summand is non-negative, if S := S 2 −ρS 1 > 0 for some positive ρ, then there is an α ∈ A(N ) such that α + h 1 , . . . , α + h k contains atleast [ρ] + 1 G K 2 -numbers. Therefore it is enough to show that S > 0 for all sufficiently large N .
Fix a δ > 0 and 0 < < δB m K , then chooseF ∈ S K so that Using Proposition 2, we obtain If ρ = m KMk B − δ, then S > 0 for large N . Since δ is arbitrary, there are infinitely many To complete the proof of the Theorem 1 it is enough to show thatr k → ∞ as k → ∞. Since the integralsĨ k (F ) in [9]). This completes the proof asr k is directly proportional toM k . k (F ) as in [9], we can show thatM k ≥ (1.0986)M k where M k is as in [9]. Since ω K = 2 for fields with class number more than 2, From Proposition 4.3 of [9], it follows that

Remark 3 Comparing the integralĨ
for sufficiently large k. We conclude that there exist infinitely many α ∈ O K such that for any admissible k-tuple (h 1 , · · · , h k ) there is atleast one G K 2 -number among α + h 1 , · · · , α + h k (i.er k ≥ 1) provided log k − 2 log log k ≥ (1.82)h K + 2 and in that case the gap is bounded above by h k − h 1 where (h 1 , · · · , h k ) is an admissible k-tuple. Therefore gaps between G K 2 -numbers are bounded in terms of class numbers.
To prove the Corollary 1 stated in Sect. 1, we need following lemmas .

Lemma 15 (Proposition 3.1, [1]) Suppose that H is an admissible tuple in Z.
Then H is also an admissible tuple in O K for every number field K .