Small Gaps Between Three Almost Primes and Almost Prime Powers
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Thus there are bounded gaps containing infinitely often ν +1 members of any of (j) (j) (j) the sequences Sn , sn , qn since any Ej -number is both a j-almost prime and a j-almost prime power. When ν = 1, we have the following results unconditionally:
lim inf(pn+1 − pn) ≤ 246, [Pol14] n→∞ (2) (2) lim inf(q − qn ) ≤ 6, [GGPY09] n→∞ n+1 (3) (3) (3) (3) lim inf(S − Sn ) ≤ lim inf(q − qn ) ≤ 2, [GGPY11] n→∞ n+1 n→∞ n+1 lim inf(s(3) − s(3))=1, [GGPY11] n→∞ n+1 n and for j ≥ 4, we get lim inf(s(j) − s(j)) = lim inf = (S(j) − S(j))=1. [GGPY11] n→∞ n+1 n n→∞ n+1 n We can can also get sharper bounds by assuming some widely believed conjectures. For example, if there are infinitely many Mersenne primes 2p − 1, then lim inf(s(1) − s(1))=1. n→∞ n+1 n If we assume the Elliott-Halberstam conjecture for primes (see Section 5), then we get a much smaller gap for primes:
lim inf(pn+1 − pn) ≤ 12. [May15] n→∞ In fact, a generalized Elliott-Halberstam conjecture can reduce this gap to 6, while the still unproven twin prime conjecture asserts that lim infn→∞(pn+1 − pn) = 2. The twin prime conjecture follows from a special case of the prime k-tuple conjecture, the first Hardy-Littlewood conjecture, which we will describe in the next section. In this paper, we are interested in the case ν = 2 and j ≥ 2. This means we want to find small gaps which contain three j-almost primes or j-almost prime powers infinitely often. For the primes themselves (i.e., when j = 1), the k-tuple conjecture for k = ν + 1 = 3 implies that
lim inf(pn+2 − pn)=6. n→∞ Unconditionally, we know
lim inf(pn+2 − pn) ≤ 398130, [Pol14] n→∞ and assuming the Elliott-Halberstam conjecture for primes we can improve this to
lim inf(pn+2 − pn) ≤ 270. [Pol14] n→∞ For three j-almost primes or j-almost prime powers with j ≥ 2, we can get much smaller gaps. For instance, assuming the Elliott-Halberstam conjecture for primes and E2-numbers, Sono proved (1) lim inf(q(2) − q(2)) ≤ 12 [Son20] n→∞ n+2 n and lim inf(r(2) − r(2)) ≤ 6 [Son20] n→∞ n+2 n (2) where rn denotes the nth P2-number (i.e., a prime or E2). SMALLGAPSBETWEENTHREEALMOSTPRIMES 3
Sono used a multi-dimensional, Maynard-Tao sieve to obtain his results. Similar techniques have been applied to Ej -numbers for j > 2 as in [LPS17]. Here we use the GGPY sieve to give an alternate proof of Sono’s inequality in Equation (1) and also prove an unconditional version which has not been previously derived. Let EH(P, E2) denote the Elliot-Halberstam conjecture for primes and E2-numbers. Theorem 1. We have lim inf(q(2) − q(2)) ≤ 32. n→∞ n+2 n
If we assume EH(P, E2), then
lim inf(q(2) − q(2)) ≤ 12. n→∞ n+2 n We use combinatorial methods which complement the sieve results to prove our main theorems. Theorem 2. We have lim inf(q(3) − q(3)) ≤ 15. n→∞ n+2 n
If we assume EH(P, E2), then
lim inf(q(3) − q(3)) ≤ 5. n→∞ n+2 n Theorem 3. We have
(4) (4) (5) (5) lim inf(s − sn ), lim inf(S − Sn ) ≤ 9. n→∞ n+2 n→∞ n+2
If we assume EH(P, E2), then
lim inf(s(3) − s(3)), lim inf(S(4) − S(4)) ≤ 4. n→∞ n+2 n n→∞ n+2 n We will also show that Theorem 3 is optimal in the sense that smaller gap sizes cannot be obtained with current methods. See Theorem 10. Let d(x) denote the number of divisors of x. Then the twin prime conjecture is equivalent to the statement d(x) = d(x + 2) = 2 for infinitely many x. The authors and their collaborators showed in [GGP+21] that for every integer n> 0, there are infinitely many x such that x and x + n have the same fixed exponent pattern, i.e., multiset of exponents in the prime factorization; in particular, d(x)= d(x + n)= c for some fixed c (depending on n) and infinitely many x. For example, both 180 = 22 · 32 · 51 and 300 = 22 · 31 · 52 have exponent pattern {2, 2, 1} with d(180) = d(300) = 18. Little is known about multiple shifts x < x + a < x + b. It is conjectured that d(x) = d(x +1) = d(x + 2) = 4 for infinitely many x, but no statement of the form d(x) = d(x + a) = d(x + b) for infinitely many x has been established for any particular a, b. Here we prove the following. Theorem 4. There are integers a,b with 1 ≤ a
2. Notation and Preliminaries Definition 1. We define a linear form to be an expression L(m)= am + b where a and b are integers with a> 0, and we say L is reduced when (a,b) = 1. We will view a linear form as both a polynomial and a function of m ∈ N. Dirichlet’s theorem on primes in arithmetic progressions guarantees that any reduced linear form will assume prime values infinitely often. In fact, if L(m) = am + b is reduced and we define π(x; a,b) = #{p ≤ x : p = L(m) is prime for some m ∈ N} then we have the asymptotic li(x) (2) π(x; a,b) ∼ ϕ(a) as x → ∞ where li(x) is the logarithmic integral and ϕ(a) denotes the Euler phi- function. Thus if we define the related prime counting function π(x; L(m)) = #{m ∈ N : m ≤ x and L(m) is prime} we get a x (3) π(x; L(m)) ∼ · ϕ(a) log x as x → ∞. The question of when two or more reduced linear forms simultaneously assume prime values infinitely often is addressed by the unproven Hardy-Littlewood prime k-tuples conjecture, which states roughly that linear forms will simultane- ously assume prime values infinitely often unless there is some obvious congruence preventing it. For example, the triple of forms m, m + 2, and m + 10 cannot be simultaneously prime infinitely often since their product is congruent to zero mod- ulo 3 for any m. To state a more precise form of the conjecture we need a few definitions. Let L denote a k-tuple of linear forms, i.e., a sequence of k distinct forms L1 = a1m + b1, L2 = a2m + b2, ..., Lk = akm + bk. Next, define a singular series for L via 1 −k ν (p) S(L)= 1 − 1 − L , p p p Yprime k where νL(p) denotes the number of solutions m ∈{1, 2,...,p} to i=1 Li(m) ≡ 0 (mod p). Note that if νL(p)= p for some prime number p, then SQ(L) = 0. When νL(p) < p, we say L is p-admissible. If L is p-admissible for all prime numbers p, then S(L) 6= 0, and in this case the k-tuple L is called admissible. Note that admissibility of L is equivalent to having all forms in L reduced and having p- admissibility for all primes p ≤ k. We define a prime counting function for the k-tuple L as
π(x; L) = #{m ∈ N : m ≤ x and Li(m) is prime for all i}. We can now state a generalization of the asymptotic in Equation (3). Conjecture (Hardy-Littlewood Prime k-Tuple Conjecture). For each fixed k ≥ 1 and admissible k-tuple L as above, we have x (4) π(x; L) ∼ S(L) · (log x)k as x → ∞. SMALLGAPSBETWEENTHREEALMOSTPRIMES 5
While a proof of the prime k-tuple conjecture appears beyond our current state of knowledge (even in the cases k = 2 or 3), we do know the following major result which was proven in [GGPY09]. Theorem 5. Let C be any constant and ν be any positive integer. Then for k sufficiently large, every admissible k-tuple of linear forms has ν +1 among them which infinitely often take E2-numbers simultaneously as values with both prime factors above C. The authors in [GGPY09] showed further that we may take k = 3 when ν = 1. This means every admissible triple contains two forms which simultaneously assume E2-numbers with large prime factors as values infinitely often. For ν = 2 we have the following. Theorem 6. Let C be any constant. Then every admissible 10-tuple of linear forms has 3 among them which infinitely often take E2-numbers simultaneously as values with both prime factors above C. If we assume EH(P, E2), then the same conclusion holds for every admissible 5-tuple. The proof of this result, which is largely computational, is postponed until Sec- tion 5, the last in the paper. However, Theorem 6 allows us to give a quick proof of Theorem 1.
Proof of Theorem 1. The 10-tuple m, m + 2, m + 6, m + 8, m + 12, m + 18, m + 20, m + 26, m + 30, m + 32 is admissible. The 5-tuple m, m + 2, m + 6, m + 8, m +12 is admissible.
We now develop the combinatorial tools needed to prove our main results, The- orems 2 to 4.
3. Relation Diagrams and Adjoining Transformations
Theorem 6 says that for k ≥ 10 (or k ≥ 5 assuming EH(P, E2)), every admissible k k-tuple L = (Li)i=1 contains 3 forms Lh, Li, Lj such that Lh(m), Li(m), Lj (m) are all E2-numbers with large prime factors for each m belonging to an infinite set M of positive integers. We can take integer combinations of these three forms to get small gaps between three almost primes. For example, if Lh = 10m + 1, Li = 15m + 2, Lj =6m +1, then 3Lh =2Li − 1=5Lj − 2, so for each m ∈ M we would get three E3 numbers 3Lh(m), 2Li(m), 5Lj(m) within a gap size of 2. The difficulty here is that Theorem 6 does not specify which triple of forms Lh, Li, Lj in our k-tuple does the job; the theorem only guarantees that such a triple exists. Thus we must find admissible k-tuples where every triple of forms contained in it gives us small gaps for three of the same kind of almost primes.
Definition 2. Given two linear forms L1 and L2, a relation from L1 to L2 is an equation of the form
c2 · L2 − c1 · L1 = r where c1, c2, and r are all positive integers. We denote such a relation by (c1)r (c2) L1 L2 6 DANIEL GOLDSTON, APOORVA PANIDAPU, AND JORDAN SCHETTLER and we call c1, c2 the relation coefficients and r the relation value. The distance between L1 and L2, denoted by dist(L1,L2), is defined to be the minimum value of r in any such relation. A relation diagram is a directed graph where the vertices are reduced linear forms and the edges are relations. Here we will always assume our diagrams are simple, i.e., have at most one edge between each pair of forms. A triangle relation for a triple of forms is a relation diagram of the form seen in Figure 1.
L (c2) 2 (c2)
r1 r2
(c1) (c3)
L1 L3 (c1)r1 + r2 (c3) Figure 1. A General Triangle Relation
The diameter of the triple, denoted by diam(L1,L2,L3), is defined to be the mini- mum value of the sum r1 + r2 in any such triangle relation.
Lemma 7. There is a strict total order on reduced forms given by Li 7−→ Lj whenever there is a relation from Li to Lj . Given a triple of linear forms Li(m)= aim + bi for i =1, 2, 3 with L1 7−→ L2 7−→ L3 we have
b2 b1 dist(L1,L2) = [a1,a2] − a2 a1 and b3 b1 diam(L1,L2,L3) = [a1,a2,a3] − a3 a1 where [a1,a2] and [a1,a2,a3] denote least common multiples. Proof. Note that since we assume linear coefficients and relation values are positive, Li 7−→ Lj is equivalent to the inequality bi/ai < bj/aj . This gives us transitivity. Trichotomy follows from this along with the assumption the forms are reduced. To prove the formula for distance, we need to minimize r = c2b2 − c1b1 = v(b2/a2 − b1/a1) subject to the condition v = c2a2 = c1a1 for positive integers c1,c2. The smallest value for v is [a1,a2]. To prove the formula for diameter, we need to minimize r1 + r2 = c3b3 − c1b1 = v(b3/a3 − b1/a1) subject to the condition v = c3a3 = c2a2 = c1a1 for positive integers c1,c2,c3. The smallest value for v is [a1,a2,a3]. k Definition 3. A relation diagram on a k-tuple L = (Li)i=1 of linear forms is called consistent if the graph is complete with edges of the form
(ci) ri,j (cj ) Li Lj so each form Li has the same relation coefficient ci in all of its edges. Triangle relations, for instance, are consistent, and the induced subgraph for any triple in a consistent diagram is a triangle relation. We say that a consistent diagram is also f-compatible for some arithmetic function f when f(ci) is constant across all i. SMALLGAPSBETWEENTHREEALMOSTPRIMES 7
Definition 4. Let f be an arithmetic function. We say f is a function on exponent patterns if f(n) = f(m) whenever m and n have the same exponent pattern. We say f is homomorphic if the following property holds: f(a)= f(b) and f(c)= f(d) with (a,c)=1=(b, d) implies f(ac)= f(bd). Remark. Note that any multiplicative or additive arithmetic function is homomor- phic. Examples of arithmetic functions which are both homomorphic and functions on exponent patterns include d (number of divisors), ω (number of distinct prime factors), Ω (number of prime factors), or the function h defined by h(n) = least positive integer having the same exponent pattern as n. Here the function h is neither additive nor multiplicative. The average exponent of n is an example of a function on exponent patterns which is not homomorphic.
Lemma 8. Suppose k ≥ 10 (or k ≥ 5 assuming EH(P, E2)) and that we have k an admissible k-tuple L = (Li)i=1 of linear forms satisfying a consistent relation diagram with edges
(ci) ri,j (cj ) Li Lj whenever i < j. If the diagram is also f-compatible for some homomorphic function f on exponent patterns, then there is a constant c and integers a, b with
rmin ≤ a
Proof. By Theorem 6, there are 3 forms Lh, Li, Lj in the admissible k-tuple L such that Lh(m), Li(m), Lj(m) are all E2-numbers for each m belonging to an infinite set M of positive integers. If we take
(5) x = chLh(m) for each m ∈ M, then
(6) x + rh,i = ciLi(m) and, since the diagram is consistent,
(7) x + rh,j = x + rh,i + ri,j = ciLi(m)+ ri,j = cjLj (m).
Also, f(ch) = f(ci) = f(cj ) since the diagram is f-compatible and f(Lh(m)) = f(Li(m)) = f(Lj (m)) is constant for all m ∈ M since f is a function on ex- ponent patterns and Lh(m), Li(m), and Lj(m) are all E2-numbers. In fact, we may assume the prime factors of Lh(m), Li(m), and Lj (m) are all larger than any of the relation coefficients in the diagram, so since f is homomorphic, we get c := f(chLh(m)) = f(ciLi(m)) = f(cj Lj(m)) is a constant across all m ∈ M. In fact, c is also independent of h,i,j by f-compatibility. The result now follows by inspection of Equations (5) to (7). Example 1. Consider the relation diagram in Figure 2. The 5-tuple is admissible and the diagram is consistent, but the diagram is not f-compatible for any f ∈ {d, ω, Ω,h}. However, all the relation coefficients are either 1 or a prime, and we show later how to “adjoin” primes to get a consistent diagram on an admissible 8 DANIEL GOLDSTON, APOORVA PANIDAPU, AND JORDAN SCHETTLER
2m +1 (3) (3)
(3) (3) 1 5
(2) (2) (2) 4 (2) 3m +2 3m +4
(2) 2 4 (2) (2) (2)
1 3 3 1
(1) (1)
(1) (1) (1) (1) 6m +5 6m +7 (1)2 (1)
Figure 2. A Consistent Diagram on an Admissible 5-Tuple tuple with all prime relation coefficients, which will yield a new diagram that is both Ω- and ω-compatible. Example 2. Consider the 6-tuple 24m+5, 90m+19, 288m+61, 33m+7, 80m+17, 108m + 23. The tuple is admissible and the distance between any two forms here is 1, yet maxh Theorem 9 (Graham’s Conjecture). Suppose m1, m2, ..., mn are distinct positive integers. Then there are i, j such that m j ≥ n. (mi,mj ) Theorem 10. Given an integer k ≥ 3 and a k-tuple of linear forms L1 7−→ L2 7−→ · · · 7−→ Lk, we have max diam(Lh,Li,Lj ) ≥ k − 1. h Proof. Write Li(m) = aim + bi so that b1/a1 < b2/a2 < ... < bk/ak. Take A = [a1,a2,...,ak] and for i ≥ 2 define bi b1 mi = A − . ai a1 Each mi is an integer and 0 Note that A/[a1,ai,aj] is a common divisor of mi, mj , so by Lemma 7 bj b1 mj diam(L1,Li,Lj ) ≥ [a1,ai,aj ] − = aj a1 A/[a1,ai,aj] m ≥ j ≥ k − 1. (mi,mj) Remark. Theorem 10 illustrates that there are fundamental restrictions for the gap sizes that we can obtain with these methods. In particular, the upper bound rmax for the gap size b in the multiple shift x, x + a, x + b of Lemma 8 satisfies rmax ≥ max diam(Lh,Li,Lj) ≥ k − 1. h (ci,j )ri,j (cj,i) Li Lj Then the following hold: k (a) For every adjoining transformation TA,B, the k-tuple TA,B(L) = (TA,B(Li))i=1 satisfies a relation diagram with edges (gici,j )ri,j (gj cj,i) TA,B(Li) TA,B(Lj ) where the gi are the adjoining factors for Li under TA,B. Thus if the diagram for L is consistent, so is the diagram for TA,B(L). (b) We have inequalities dist(TA,B(Li),TA,B(Lj )) ≤ dist(Li,Lj) diam(TA,B(Lh),TA,B(Li),TA,B(Lj )) ≤ diam(Lh,Li,Lj) (c) If L is admissible, then for every choice of positive integers g1, g2, ..., gk such that (gi,ai) = (gi,aibj − aj bi) = (gi,gj)=1 whenever i 6= j, there is an adjoining transformation TA,B such that TA,B(L) is admissible and gi is the adjoining factor for each Li under TA,B. 10 DANIEL GOLDSTON, APOORVA PANIDAPU, AND JORDAN SCHETTLER 70m +1 (3) (3) (3) (3) 1 5 (2) (2) (2) 4 (2) 105m +2 105m +4 (2) 2 4 (2) (2) (2) 1 3 3 1 (5) (7) (5) (5) (7) (7) 42m +1 30m +1 (5)2 (7) Figure 3. An ω- and Ω-Compatible Relation Diagram Proof. Part (a) follows from definitions since cj,iLj(m) − ci,j Li(m) = ri,j implies cj,igj (Lj(Am + B)/gj ) − ci,j gi(Li(Am + B)/gi)= ri,j . Part (b) follows from part (a) combined with Lemma 7. The authors and their collaborators proved part (c) + 2 in [GGP 21]. In fact, it was shown that we can take A = (g1g2 ··· gk) and B to 2 be a solution of the congruences Li(B) ≡ gi (mod gi ). 4. Proofs of Theorems 2 to 4 In this section we use Lemmas 8 and 11 to establish proofs for our main results. Proof of Theorem 2. First, we assume EH(P, E2). Start with the admissible 5-tuple L from Example 1: 2m+1, 3m+2, 6m+5, 6m+7, 3m+4. Recall that the diagram on L in Figure 2 is consistent with relation values ranging from 1 to 5 and relation coefficients c1 = 3, c2 = 2, c3 = 1, c4 = 1, c5 = 2. We can apply a transformation to ′ ′ ′ get a new admissible tuple T35,0(L): L1 = 70m + 1, L2 = 105m + 2, L3 = 42m + 1, ′ ′ L4 = 30m + 1, L5 = 105m + 4. The associated consistent diagram is both ω- and ′ ′ ′ Ω-compatible since all relation coefficients are now prime: c1 = 3, c2 = 2, c3 = 5, ′ ′ c4 = 7, c5 = 2. See Figure 3. Thus Lemma 8 implies that there are integers a,b with 1 ≤ a lim inf(q(3) − q(3)) ≤ 5. n→∞ n+2 n To get the unconditional result about small gaps between three E3-numbers, we begin with a non-admissible 10-tuple L of monic forms: m + 4, m + 5, m + 7, m + 8, m + 9, m + 11, m + 13, m + 16, m + 17, m + 19. The trivial diagram on L with all relation coefficients equal to 1 is consistent with relation values ranging from 1 to 15. SMALLGAPSBETWEENTHREEALMOSTPRIMES 11 We apply a transformation making use of a primorial 19# = 2·3·5·7·11·13·17·19 = ′ 9699690 to get an admissible tuple L = T19#,0(L): ′ ′ L1 = 4849845m +2,L2 = 1939938m +1, ′ ′ L3 = 1385670m +1,L4 = 4849845m +4, ′ ′ L5 = 3233230m +3,L6 = 881790m +1, ′ ′ L7 = 746130m +1,L8 = 4849845m +8, ′ ′ L9 = 570570m +1,L10 = 510510m +1. The associated consistent diagram is both ω- and Ω-compatible since all relation ′ ′ ′ ′ ′ ′ ′ coefficients are prime: c1 = 2, c2 = 5, c3 = 7, c4 = 2, c5 = 3, c6 = 11, c7 = 13, ′ ′ ′ c8 = 2, c9 = 17, c10 = 19. Thus Lemma 8 implies that there are integers a,b with 1 ≤ a lim inf(q(3) − q(3)) ≤ 15. n→∞ n+2 n Proof of Theorems 3 and 4. We begin with the non-admissible monic k-tuple L: Li = m + i for i =1,...,k. The trivial diagram on L with all relation coefficients equal to 1 is consistent with relation values ranging from 1 to k − 1. We apply a transformation making use of the least common multiple A(k) = [1, 2,...,k] to get ′ an admissible tuple L = TA(k),0(L) which satisfies a consistent diagram with edges (i)j − i (j) ′ ′ Li Lj whenever i < j. We will adjoin factors gi by Lemma 11 in various ways to maintain ′ admissibility and get relation coefficients ci = igi in our consistent diagram. Note ′ ′ ′ ′ ′ that the linear coefficients ai = A(k)/i and the determinants aibj −ajbi = A(k)(j − i)/(ij) are only divisible by primes less than or equal to k, so we take our gi to consist of primes greater than k. First we assume EH(P, E2), so we may take k =5 here by Theorem 6. Now adjoin g1 = 7 and gi = 1 otherwise to get an admissible ′ 5-tuple T72,5(L ): 420m+43, 1470m+151, 980m+101, 735m+76, 588m+61. The associated consistent diagram is ω-compatible with relation coefficients satisfying ω(igi) = 1 for all i, so Lemma 8 implies there are integers a,b with 1 ≤ a (3) (3) lim inf(s − sn ) ≤ 4. n→∞ n+2 Next, we adjoin g1 = 7 · 11, g2 = 13, g3 = 17, g4 = 1, and g5 = 19 to get an ′ 2 admissible 5-tuple TA,B(L ) with A = 323323 and B = 97650202718: 81457996620m + 76091067053, 241240989990m + 225346621657, 122985602740m + 114882591433, 1568066434935m + 1464753040771, 66023849892m + 61673812243. 12 DANIEL GOLDSTON, APOORVA PANIDAPU, AND JORDAN SCHETTLER The associated consistent diagram is Ω-compatible with relation coefficients satisfy- ing Ω(igi) = 2 for all i, so Lemma 8 implies there are integers a,b with 1 ≤ a 5. The GGPY Sieve and Proof of Theorem 6 In this last section, we use the GGPY sieve of [GGPY09] to prove Theorem 6. This will complete the proofs of Theorems 1 to 4. We seek to establish when there SMALLGAPSBETWEENTHREEALMOSTPRIMES 13 are three sifted E2-numbers simultaneously infinitely often in an admissible k-tuple. The conjectural versus conditional results are distinguished by a common level of distribution for primes and E2-numbers. Definition 6. Define an error E(x; a,b)= π(x; a,b)−li(x)/ϕ(a) for the asymptotic in Equation (2). We say the primes have a level of distribution ϑ if for all A > 0 there is a constant C (depending on A) such that x max |E(x; a,b)| ≪A (a,b)=1 (log x)A a≤xϑX/(log x)C We can define a level of distribution for E2-numbers in a completely analogous way. Theorem 12 (Bombieri-Vinogradov, [Bom87], [Vin65]). The primes have a level of distribution 1/2. Theorem 13 (Motohashi, [Mot76]). The E2-numbers have a level of distribution 1/2. Friedlander and Granville [FG89] showed that the primes do not have a level of distribution 1. However, the following conjecture, which first appeared for primes in [EH70], may still hold. Conjecture (Elliot-Halberstam, EH(P, E2)). Primes and E2-numbers have a com- mon level of distribution ϑ for every ϑ< 1. We define an indicator function for sifted E2-numbers as follows. Take β(n)=1 η 1/2 if n = p1p2 where each pi is a prime with N 1 2 k−1 J0 = P (1 − x) x dx, Z0 1 1−y 2 B k−2 J1 = P˜(1 − x) − P˜(1 − x − y) x dx dy, ZBη y(B − y) Z0 1 1 B 2 k−2 J2 = P˜(1 − x) x dx dy, ZBη y(B − y) Z1−y B/2 1 B 2 k−2 J3 = P˜(1 − x) x dx dy. Z1 y(B − y) Z0 Here B =2/ϑ where ϑ is a common level of distribution for primes and E2-numbers. Thus to get small gaps between ν + 1 = 3 sifted E2-numbers, we need to show that J > 0 for some choices of B (either B = 4 unconditionally or B =2+ ǫ assuming EH(P, E2)), η ∈ (0, 1/4], and some polynomial P (x). Proof of Theorem 6. For ν = 2 and B = 4, we can take k = 10, η = 1/340, and 3 3 2 P (x)= 20 + 5 x + 10x . Then 18549 J0 = = 0.02316308 ..., 800800 2113287710672781837420508478315754592524 1262566905669 113 J1 = − log 105076848611852709873077392578125 17875 85 = 0.00063269 ..., 79584691575671328932238080587887183154 12980396496724 113 J2 = − + log 3967936940587444988214111328125 184275 85 = 0.00074896 ..., 911 J3 = log(3) = 0.00067890 ..., 1474200 8719967520249406350967107646046792519503 1953628194503 113 J = − log 7061164226716502103470800781250000 450450 85 911 + log(3) = 0.00003645 ...> 0. 65520 Now assume EH(P, E2). For ν = 2 and B = 201/100 > 2, we can take k = 5, 3 2 η =1/340, and P (x)= 4 +6x + 10x . Then 7487 J0 = = 1.48551587 ..., 5040 1185867362212062499391309326339523040406768636033 J1 = − 6426286514402549760000000000000000000000000000 75466092079924833449781 68139 + log = 0.15294205 ..., 280000000000000000000 34340 42290186710920567072992744924095611611325091277 J2 = 229510232657233920000000000000000000000000000 18808406922710397486573 68139 − log = 0.14486877 ..., 70000000000000000000 34340 1747 J3 = log (101/100) = 0.00143710 ..., 12096 54641056436520615648365200967849481935314631 218375 101 J = − + log 9639429771603824640000000000000000000000000 151956 100 1156539249170365689 68139 + log = 0.00655959 ...> 0. 140000000000000000 34340 SMALLGAPSBETWEENTHREEALMOSTPRIMES 15 References [Bom87] Enrico Bombieri, Le grand crible dans la th´eorie analytique des nombres, Ast´erisque (1987), no. 18, 103. [BS96] R. Balasubramanian and K. Soundararajan, On a conjecture of R. L. Graham, Acta Arith. 75 (1996), no. 1, 1–38. [EH70] P. D. T. A. Elliottand H. 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