# Bounded gaps between products of distinct primes

- Yang Liu
^{1}, - Peter S. Park
^{2}and - Zhuo Qun Song
^{2}Email author

**Received: **14 July 2016

**Accepted: **10 July 2017

**Published: **1 December 2017

## Abstract

Let \(r \ge 2\) be an integer. We adapt the Maynard–Tao sieve to produce the asymptotically best-known bounded gaps between products of *r* distinct primes. Our result applies to positive-density subsets of the primes that satisfy certain equidistribution conditions. This improves on the work of Thorne and Sono.

## 1 Introduction

*i*-th prime. The numerical evidence for this conjecture is striking because the average value of \(p_{n+1} - p_n\) is known to grow arbitrarily large; specifically, the prime number theorem implies that the difference is \({\sim }{\log p_n}\) on average. One of the first significant advances toward the twin prime conjecture is due to Chen Jingrun [2], who proved in 1973 that there are infinitely many primes

*p*such that \(p+2\) is a product of at most two primes.

*level of distribution*if (1.2) holds for any \(A>0\). Also, these methods conditionally prove bounded gaps between the primes, i.e, a finite bound for

*r*distinct primes in \(\mathcal P\). GGPY [6] proved that

*i*-th number in \(E_2(\mathbb P)\). Subsequently, Thorne [14] extended their method to prove the existence of bounded gaps between any number of consecutive \(E_r(\mathbb P)\) numbers for any \(r \ge 2\). In fact, Thorne’s result extends to \(E_r(\mathcal P)\) for subsets \(\mathcal P\) of the primes satisfying certain equidistribution conditions, thus yielding a number of intriguing consequences in the context of multiplicative number theory. For instance, for an elliptic curve \(E/\mathbb Q\) given by the equation

*D*, let

*D*-quadratic twist of

*E*. Thorne proved in [14, Theorem 1.2] that if

*E*has no 2-torsion, then there are bounded gaps between squarefree fundamental discriminants for which

*E*(

*D*) has Mordell–Weil rank 0 and its Hasse–Weil

*L*-function satisfies \(L(1,E(D)) \ne 0\). Another corollary, [14, Corollary 1.3], states that there are bounded gaps between \(E_2(\mathbb P)\) numbers

*pq*such that the class group \({\text {Cl}}(\sqrt{-pq})\) contains an element of order 4. In 2014, Chung and Li [3] proved an analogous and quantitatively stronger bound for the size of the gaps in the above results for squarefree numbers whose prime divisors are all in \(\mathcal {P}\) instead than \(E_r(\mathcal {P})\) numbers.

*any*level of distribution \(\theta > 0\). This suggests that the Maynard–Tao sieve can be adapted to study gaps between primes in special subsets. In particular, Thorner [15] extended the methods in [8] to show that there exist bounded gaps between primes in Chebotarev sets. There are a number of interesting number-theoretic consequences for ranks of elliptic curves, Fourier coefficients of modular forms, and primes represented by binary quadratic forms.

*m*in terms of \(\varepsilon \). Moreover, Neshime [12, Remark 7.1] was able to modify Sono’s methods to show an even stronger bound of

*Siegel–Walfisz condition*if there exists a squarefree positive integer \(B = B(\mathcal {P})\) such that for all \((q, B) = 1\) and \((a, q) = 1\),

*level of distribution*\(\theta > 0\) if

*A*. For the purposes of our paper, assume that \(\theta < \frac{1}{2}\) throughout.

*admissible*if for every prime \(\ell \), there exists an integer \(n_\ell \) such that

*m*\(E_r(\mathcal P)\) numbers, for infinitely many

*n*.

### Theorem 1.1

Suppose that a subset \(\mathcal {P}\) of the primes has positive density \(\delta = \delta (\mathcal P)\) in the set of all primes, satisfies a Siegel–Walfisz condition as in (1.5), and has a positive level of distribution as in (1.6), with \(B = B(\mathcal P)\) defined accordingly. There exists a constant \(C(r,m,\mathcal P)\), depending only on *r*, *m*, and \(\mathcal P\), such that for any admissible set of linear forms \(\{n + h_1, \ldots , n + h_k\}\) satisfying \(k > C(r, m, \mathcal {P})\), it holds for infinitely many positive integers *n* that at least *m* of \(n + h_1, \ldots , n + h_k\) are \(E_r(\mathcal {P})\) numbers.

Note that the explicit value for the constant \(C(r,m,\mathcal P)\) is given in (2.1).

From Theorem 1.1, we obtain bounded gaps for an arbitrary number of \(E_r(\mathcal P)\) numbers.

### Corollary 1.2

*i*-th \(E_r(\mathcal P)\) number. Then, there exists a constant \(C(r,m,\mathcal P)\), depending only on

*r*,

*m*, and \(\mathcal P\), such that

The explicit value for the constant \(C(r,m,\mathcal P)\) of this context is given in (2.2).

When *r* is fixed and *m* is large, our result improves on the work of [14], thus yielding improved effective constants in the contexts described in [14, Section 1].

### Corollary 1.3

Let \(K/\mathbb Q\) be a Galois extension, and \(r \ge 2\) be an integer. Then, there exists an effectively computable constant *C*(*r*, *m*, *K*) such that there are infinitely many nonconjugate *m*-tuples of ideals \(\mathfrak a_1,\ldots ,\mathfrak a_m\) whose norms are \(E_r(\mathbb P)\) numbers that are simultaneously contained in an interval of length *C*(*r*, *m*, *K*).

### Proof

Apply Theorem 1.2 to the Chebotarev set of primes that are inert in *K*. \(\square \)

### Corollary 1.4

*C*(

*r*,

*m*,

*E*) such that there are infinitely many

*m*-tuples of \(E_{2r}(\mathbb P)\) numbers \(b_1< \cdots < b_{m}\) for which

- (1)
\(L\left( 1,E\left( N_E \cdot b_i\right) \right) \ne 0\) for all \(1 \le i \le m\),

- (2)
\({\text {rank}}E\left( N_E \cdot b_i\right) = 0\) for all \(1 \le i \le m\), and

- (3)
\(b_{m}-b_1< C(r,m,E)\).

### Proof

For each \(E/\mathbb Q\), one can apply Theorem 1.2 to a proof analogous to that of [14, Section 6], which computes the constants for the elliptic curve \(X_0(11):y^2 =x^3-4x^2-160x-1264\). The computations rely on the work of Ono [11] and the Bombieri–Vinogradov-type result of Murty and Murty [10]. \(\square \)

### Corollary 1.5

*C*(

*r*,

*m*) such that there are infinitely many

*m*-tuples of \(E_r(\mathbb P)\) numbers \(d_1< \cdots < d_m\) such that the following hold:

- (1)
The class group \({\text {Cl}}\left( \mathbb Q(\sqrt{-d_i})\right) \) has an element of order 4 for all \(1 \le i \le m\).

- (2)
\(d_m-d_1< C(r,m)\).

### Proof

Soundararajan [13] has proved that for any squarefree \(d \equiv 1 \pmod 8\) whose prime factors are congruent to either 1 or \(-1 \pmod 8\), the class group \({\text {Cl}}\left( \mathbb Q(\sqrt{-d})\right) \) contains an element of order 4. Our corollary then follows immediately from applying Theorem 1.2 to \(\mathcal P = \{p \in \mathbb P:p \equiv 1 \pmod 8\}\). \(\square \)

In [14], Thorne proved in the contexts of Corollaries 1.3, 1.4, and 1.5, we can take the constants \(C(r, m, K) = C(r, K) \exp \left( m^{\frac{1}{r-2}}\right) \), \(C(r, m, E) = C(r, E) \exp \left( m^{\frac{1}{r-2}}\right) \), and \(C(r, m) = C(r) \exp \left( m^{\frac{1}{r-2}}\right) \), respectively. When *r* is fixed and *m* is large, we improve these bounds to \(C(r, m, K) = C(r, K) m^{\frac{1}{r}} \exp \left( m^{\frac{1}{r}}\right) \), \(C(r, m, E) = C(r, E) m^{\frac{1}{r}} \exp \left( m^{\frac{1}{r}}\right) \), and \(C(r, m) = C(r) m^{\frac{1}{r}} \exp \left( m^{\frac{1}{r}}\right) \), respectively. We note that these bounds are not as strong as those obtained by Chung and Li in [3] when considering square-free numbers with no restriction on the number of prime factors.

In Sect. 2, we list the definitions and notations we will be using, state a precise version of our main result, and give an outline of our proof. Then, in Sect. 3, we prove a version of the Bombieri–Vinogradov theorem for \(E_r(\mathcal P)\) numbers. We then use this result to prove our main theorem in the course of Sects. 4, 5, 6, and 7. Finally, in Sect. 8, we explicitly compute the constant given in Corollary 1.5 for \(r=2\) and arbitrary *m*, which shows a concrete application of our asymptotically improved result on bounded gaps.

## 2 Preliminaries

We detail below the equidistribution properties that we will assume for our infinite subset \(\mathcal {P} \subset \mathbb P\), for which we will show that the gap between *m* consecutive \(E_r(\mathcal P)\) is bounded infinitely often.

### 2.1 Definitions and notations

All sums, products, and maxima are taken with the variables ranging over the positive integers \(\mathbb N\). The exception to this will be that variables denoted by *p* or \(p_i\) will be assumed to be taken over \(\mathcal P\), and the variable \(\ell \) will be assumed to be taken over \(\mathbb P\). For the purposes of our paper, the level of distribution \(\theta \) is assumed to be less than 1 / 2. Throughout the proof, we work with a fixed admissible set of *k* distinct linear forms \(\mathcal H\), and without loss of generality assume that \(h_1< \cdots < h_k\).

We let \(\phi \) denote the Euler’s totient function, \(\mu \) the Möbius function, and \(\tau _u\) the function given by the number of distinct ways a number can be written as a product of *u* ordered positive integers. For two positive integers *a*, *b*, we let (*a*, *b*) denote their greatest common divisor and [*a*, *b*] their least common multiple. The exception to this is in Sect. 3, where [*a*, *b*] will denote the closed interval with endpoints *a* and *b*.

*N*, we further restrict our consideration to products of

*r*distinct primes in \(\mathcal P\) such that the prime factors satisfy a size constraint in terms of

*N*. Specifically, for every \(1 \le h \le r\), let \(E_h := \{p_1 \ldots p_h :N^\eta \le p_1< \cdots < p_h \text { and } N^{\frac{1}{2}} \le p_h\}\), where \(0< \eta < \frac{1}{r}\) is fixed. We note that the definition of \(E_h\) has an implicit dependence on

*N*. Furthermore, let \(\beta _h\) be the indicator function for \(E_h\), i.e,

*o*,

*O*, and \(\ll \), are to be interpreted as \(N \rightarrow \infty \); in particular, the implied constants may depend on \(k, r, \eta ,\) and \(\mathcal H\). We will in multiple instances let \(\varepsilon , \epsilon ,\) and \(\epsilon _1\) denote positive real numbers that one can take to be sufficiently small.

### 2.2 Statement of precise results

We now state a precise version of our main theorem.

### Theorem 2.1

*n*, at least

*m*of \(n+h_1, \dots , n+h_k\) are \(E_r(\mathcal {P})\) numbers, given that

### Remark

For \(\mathcal P = \mathbb P\), we have \(\delta = 1\), \(B=1\), and \(\theta _1 = \frac{1}{2}- \varepsilon \) for all small \(\varepsilon \).

Given this theorem, we easily obtain a corollary about bounded gaps between \(E_r(\mathcal {P})\) numbers.

### Corollary 2.2

*n*-th \(E_{r}(\mathcal {P})\) number,

### Proof

In Theorem 2.1, we can let \(h_i\) be the *i*-th prime greater than *k* for \(1 \le i \le k.\) It is easy to verify that \(\{n + h_1, n + h_2, \ldots , n + h_k\}\) is an admissible tuple of linear forms, and that \(h_k - h_1 \ll \mathcal {L} \log \mathcal {L}.\)
\(\square \)

### 2.3 Outline of proof

*N*, \(B \mid W\), so we can define \(U = \frac{W}{B}\). For each prime \(\ell \mid U\), choose a residue class \(v_{\ell } \pmod \ell \) such that

*U*such that

*n*must be positive; for this value of

*n*, at least \(\left\lfloor \rho \right\rfloor + 1\) of \(n+h_1, \ldots , n+h_k\) are in \(E_r\), as desired. It remains to show that for some choice of \(w_n\), \(S(N, \rho ) > 0\) for all sufficiently large

*N*. Let \(R := N^{\frac{\theta }{2} - \epsilon }\). We will choose \(w_n\) to be the Maynard–Tao weights:

*N*for some choice of \(\lambda _{d_1,\ldots ,d_k}\).

### Proposition 2.3

*F*be a smooth function supported on

The proof of Proposition 2.3 depends on a Bombieri–Vinogradov-type result, proved in Sect. 3:

### Theorem 2.4

*u*and

*A*.

(2.5) was proved in [15]. We will prove (2.6) in Sects. 4, 5, and 6.

### Proposition 2.5

*k*distinct linear forms given by \(n+h_1, \ldots , n+h_k\). Let \(\mathcal {S}_{k,\eta }\) denote the space of smooth functions \(F :[0,1]^k \rightarrow \mathbb R\) supported on \(\mathcal {R}_{k,\eta }\), with \(I_k(F) \ne 0 \) and \(J_k^{(m)}(F)\ne 0\) for \(1 \le m \le k\). Define

### Proof

*N*. Thus, there are infinitely many \(n \in \mathbb N\) such that at least \(\left\lfloor \rho \right\rfloor +1\) of the numbers \(n+h_i\) are in \(E_r\). Since \(\left\lfloor \rho \right\rfloor +1 = \nu \) for sufficiently small \(\varepsilon \), taking \(\varepsilon \rightarrow 0\) (and accordingly, \(\epsilon , \epsilon _1 \rightarrow 0\)) gives the result. \(\square \)

*k*. In Section 7, we make our choice of \(\eta \) and \(F\in \mathcal {S}_{k,\eta }\) so that

## 3 Proof of Bombieri–Vinogradov for products of primes

In [14], a Bombieri–Vinogradov result was proved for \(E_r(\mathcal {P})\) numbers. The proof uses the following result of Bombieri, Friedlander, and Iwaniec [1, Equation 1.5], which was inspired by the work of Motohashi [9].

*convolution*of two arithmetic functions

*a*and

*b*, denoted \(a * b\), is defined as

### Lemma 3.1

*N*. Assume that \(\{b(i)\}\) satisfies the following condition for any \((d, k) = 1\), \((l, k) = 1\): for any constant \(A > 0\),

For our purposes, we will require a Bombieri–Vinogradov theorem for those \(E_r(\mathcal {P})\) numbers with prime factors restricted to certain intervals.

### Theorem 3.2

*u*,

*A*, and \(\varepsilon \).

*c*values of \(i \le r\). Define \(\Delta _r := \frac{1}{c} \beta _{r-1}*I_{\mathcal {P}_r} - \beta _r\). We can approximate \(\beta _r\) with \(\beta _{r-1} * I_{\mathcal {P}_r}\) using the Triangle Inequality as follows:

*q*. For every number

*p*in \([N^{a_r}, N^{b_r}]\), at most \(\frac{N^u}{p^2}\) numbers less than \(N^u\) are divisible by \(p^2\). Thus, for each

*q*, we have

*q*, we obtain

*p*in \([N^{a_r}, N^{b_r}]\), at most \(\frac{N^u}{p^2q} + 1\) numbers less than \(N^u\) are divisible by \(p^2\) and are congruent to \(a \pmod {q}\). Thus, a similar approach gives

*n*], \(J_{\omega } := [2^{\omega }, 2^{\omega + 1})\). For each \(J_{\omega }\), define the interval \(K_{\omega } = [1, \frac{N^u}{2^{\omega }}]\). Let the restriction of \(\beta _{r-1}(x)\) and \(I_{\mathcal {P}_r}\) to the intervals \(J_\omega \) and \(K_\omega \) be \(\beta _{r-1, \omega }(x)\) and \(I_{\mathcal {P}_r, \omega }\), respectively. Note that

## 4 Reduction to counting prime numbers

We reduce \(S_2^{(m)}\) to the following sum involving the prime counting function \(X_n\).

### Lemma 4.1

### Proof

*N*, the inner sum vanishes if \(U, [d_1, e_1] , \ldots , [d_k,e_k]\) are not pairwise coprime. Indeed, if \(p \mid U\) and \(p \mid [d_i, e_i]\), then either \(\lambda _{d_1, \dots , d_k} = 0\) or \(\lambda _{e_1, \dots , e_k} = 0\), as \(\lambda _{d_1, \dots , d_k}\) is supported only when \(\mu ^2(BU \prod _{i = 1}^m d_i) = 1.\) Now assume \(p \not \mid W\). If \(p\mid [d_i, e_i]\) and \(p \mid [d_j, e_j]\), then \(p \mid n+h_i\) and \(p \mid n+h_j.\) This implies that \(p \mid h_i-h_j\), which contradicts \(D_0 > h_k-h_1\) (as \(D_0\) grows with

*N*). Thus, by the Chinese remainder theorem,where \(\sum ^{'}\) denotes the restriction that \(U, [d_1, e_1] , \ldots , [d_k,e_k]\) are pairwise coprime.

*U*) and (mod \([d_i,e_i]\)) for \(i\ne m\) into a single congruence \(v \pmod {U \prod _{i \ne m}[d_i,e_i]}\). For

*n*such that \([d_m, e_m] \mid n+h_m\), we can make the change of variables \(t = \frac{n+h_m}{[d_m,e_m]}\) to obtain

*t*such that \(\beta _h(t) = 1\) but \(\beta _r(n+h_m)=0\) for the corresponding

*n*. The latter occurs precisely when at least one prime dividing \([d_m, e_m]\) also divides

*t*, since then the corresponding \(n + h_m\) is not the product of distinct primes. We bound the error bywhere we have used the fact that \(U\prod _{i \ne m}[d_i,e_i] \le R^2U\), since \(\prod _{i=1}^m d_i, \prod _{i=1}^m e_i < R\). Also, the fact that \(p \mid [d_m,e_m]\) implies that \(p \not \mid U\prod _{i \ne m} [d_i,e_i]\), which allows us to conclude that

*N*and thus

*W*. Thus, observing that \(\frac{N}{[d_m, e_m]}\) and \(\frac{2N}{[d_m, e_m]}\) are expressible as \(N^u\) for some \(u\in \mathbb R\), we can apply Theorem 3.2, as desired. Thus, we have for \(1 \le h \le r-1\) that

*t*in the range \(\frac{N+h_m}{[d_m,e_m]} \le t < \frac{2N+h_m}{[d_m,e_m]}\) such that \(\beta _r(t) = 1\). If the prime factors of

*t*that are less than or equal to \(N^{\frac{1}{2}}\) are all distinct from the prime factors of \([d_m,e_m]\), we can make a change of variables that divides

*t*by these prime factors to obtain its largest prime factor

*u*, obtaining the sumWe can bound the error resulting from repeated prime factors dividing both \([d_m,e_m]\) and

*t*as follows:Just as before, the error term resulting from \(\frac{N}{N^\eta [d_m,e_m]}\) is \(\ll \lambda _{\max }^2N^{1-\eta +\varepsilon }\), and the error term resulting from the

*O*(1) is \(\ll \lambda _{\max }^2 R^2N^{\varepsilon }\).

*h*, we obtainSince

## 5 Combinatorial sieve manipulations

### Lemma 5.1

*g*denotes the totally multiplicative function defined on the primes by \(g(\ell ) = \ell -2\). Then, we havewhere

Note that \(y_{r_1, \ldots , r_k}^{(m, q)}\) is supported on \((r_1, \ldots , r_k)\) such that \(r_m \mid q.\)

### Proof

*u*and

*d*variables gives that the previous quantity is equal toLet us now consider the conditions that we have imposed from the \('\) in the second summation. We required that \((W, d_i) = 1, (W, e_i) = 1\) and \((d_i, e_j) = 1\) for \(i \ne j\). Note that the first two conditions can be removed due to the restriction of the support of \(\lambda _{d_1,\ldots ,d_k}\). We remove the last condition \((d_i, e_j) = 1\) by introducing variables \(s_{i, j}\) for \(i \ne j\). Since

*s*and

*d*variables gives us

*d*and

*e*variables completely, we can substitute in the \(y^{(m, q)}\) variables to rewrite the previous expression aswhere

*q*. Thus, \(s_{m, j}\) would only have factors of size at least \(N^\eta .\) First, let us consider the case where \(s_{i, j} > D_0\), and \(i \ne m, j \ne m.\) The contribution from these terms is bounded by

*q*has \(2^{r-1}\) factors, and thus

*q*are at least \(N^\eta .\) The first error term dominates. By simplifying the main term, where \(s_{i, j} = 1\) for all

*i*,

*j*, we see that \(\square \)

In order to relate \(S_2^{(m)}\) to \(S_1\), we now express the \(y^{(m, q)}\) variables in terms of the *y* variables.

### Lemma 5.2

### Proof

*d*and

*a*variables to give

*q*has at most \(2^r\) factors and \(r_m \mid q\), and thus

## 6 Smooth choice of *y*

*F*to be chosen in Section 7. Note that this implies that \(y_{\max } \le F_{\max }\). Additionally, by Lemma 5.2, we can show that

We will use the following lemma to estimate the sum \(S_2^{(m)}\) by an integral.

### Lemma 6.1

*h*be the totally multiplicative function defined on the primes by \(h(\ell ) = \gamma (\ell )/(\ell -\gamma (\ell )).\) Let

*G*be a smooth function from \([0, 1] \rightarrow \mathbb {R}\), and let \(G_{\max } = \max _{t \in [0, 1]} |G(t)|+|G'(t)|.\) Then for any positive integer

*n*,

*G*or

*L*, and

Our main claim of this section is expressed in the following lemma.

### Lemma 6.2

Before proving Lemma 6.2, we first prove the following claim, which expresses the \(y^{(m, q)}\) variables in terms of the function *F*.

### Lemma 6.3

### Proof

We are now ready to prove Lemma 6.2.

### Proof

*p*greater than \(D_0\), we observe that the contribution from terms with \((r_i, r_j) > 1\) is

*q*in \(\sum ^{\dagger }\) only has prime factors that are at least \(N^\eta \), we know that every divisor \(d > 1\) of

*q*is at least \(N^\eta .\) Therefore, if

*F*is chosen to be supported on \(x_i \le \frac{\log N^{\eta }}{\log R}\), then all terms with \(d > 1\) in (6.2) vanish. Since \(\frac{\log N^{\eta }}{\log R} \ge \frac{2\eta }{\theta }\), we observe that imposing

*F*to be supported on

### Corollary 6.4

*F*is supported on (6.5), then

## 7 Smooth choice of *F*

We now choose a constant \(\eta \) and a function *F* to give a large lower bound for (2.8).

### Lemma 7.1

### Proof

*F*defined above is supported on \(\{(x_1, \ldots , x_k) :x_i \le T/k \}\). Since we need

*F*to be supported on (6.5) in order to apply Corollary 6.4, we let

*F*given in Lemma 6.2, we have

### Lemma 7.2

### Proof

*k*, we have

## 8 An example

*C*(

*m*) such that there are infinitely many

*m*-tuples of \(E_2\) numbers \(d_1 \le \cdots \le d_m\) such that the following hold:

- (1)
The class group \({\text {Cl}}\left( \mathbb Q(\sqrt{-d_i})\right) \) has an element of order 4 for all \(1 \le i \le m\).

- (2)
\(d_m-d_1< C(m)\).

*C*(

*m*) to be the diameter of any admissible set of size

*k*by Theorem 2.1. We choose the admissible set to consist of the elements \(n + h_1, n+h_2, \dots , n+h_k\), where \(h_i\) is the

*i*-th smallest prime greater than

*k*. Using the bounds provided in [4] as well as \(k > \exp (18)\), we have that there are at most \(\frac{k}{\log k} + \frac{k}{(\log k)^2}\) primes less than

*k*, and that the \(\lfloor k + \frac{k}{\log k} + \frac{k}{(\log k)^2} \rfloor \)-th prime is at most

## Declarations

### Acknowledgements

This research was supervised by Ken Ono at the Emory University Mathematics REU and was supported by the National Science Foundation (Grant Number DMS-1557960). We would like to thank Ken Ono and Jesse Thorner for offering their advice and guidance and for providing many helpful discussions and valuable suggestions on the paper. We would also like to thank the anonymous referees for their helpful comments and suggestions.

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## Authors’ Affiliations

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