Variants of the Selberg sieve, and bounded intervals containing many primes

1 Introduction

• Theorem 1.4 gives unconditional upper bounds of H1 ≤ 246, H2 ≤ 398 130, and H3 ≤ 24 797 814.Source: §1 · p. 2
• For large m, the paper proves Hm ≤ C m exp((4 − 28/157)m) unconditionally and Hm ≤ C m e^2m under the Elliott-Halberstam conjecture.Source: §1 · pp. 2–3
• Under the generalized Elliott-Halberstam conjecture, the paper obtains H1 ≤ 6, H2 ≤ 252, and improved explicit bounds for H3–H5.The corresponding bounds are H3 ≤ 52 116, H4 ≤ 474 266, and H5 ≤ 4 137 854.Source: §1 · pp. 2–3
• The parity problem shows that purely sieve-theoretic methods cannot improve the generalized Elliott-Halberstam bound H1 ≤ 6.The paper also notes that it does not improve Maynard’s H1 ≤ 12 using only the Elliott–Halberstam conjecture.Source: §1 · p. 3
• The unconditional H1 bound uses only Bombieri–Vinogradov, whereas the remaining stated improvements use the Elliott–Halberstam conjecture or a generalization.Source: §1 · p. 3
• Assuming the generalized Elliott-Halberstam conjecture, at least one of the twin-prime conjecture or a near-miss to even Goldbach holds.For sufficiently large multiples of six, one of n and n−2, and similarly one of n and n+2, is expressible as a sum of two primes.Source: §1 · p. 3
• The main technical advances are a broader multidimensional Selberg sieve and efficient numerical methods for its variational problems.The sieve cutoff function may have larger support, while the numerical work addresses both small dimensions and large-k asymptotics.Source: §1 · p. 3

1.1 Organization of the paper

• The paper’s organization derives distributional estimates, sieve asymptotics, variational criteria, and numerical bounds before combining them into Theorem 1.4.Source: §1.1 · p. 4

Analytic framework

• The paper uses distributional estimates for the von Mangoldt function and other arithmetic functions, relying on existing literature rather than proving new estimates.The discrepancy framework measures distribution in primitive residue classes, while Vaughan’s identity decomposes Λ into convolution terms controlled by GEHrϑs.Source: §2 · pp. 5, 7

Elliott–Halberstam estimates

• The Bombieri–Vinogradov theorem establishes EHrϑs for every fixed 0 < ϑ < 1/2, while Elliott–Halberstam conjectures this for all 0 < ϑ < 1.An estimate EHrϑs with some fixed ϑ > 1/2 would imply finiteness of H1, but this remains unproved.Source: §2 · p. 6

Motohashi–Pintz–Zhang estimates

• The previous-project estimate MPZrϖ, δs holds whenever 600ϖ + 180δ < 7, allowing ϖ to approach 7/600 ≈ 0.01167.This estimate concerns smooth-modulus distribution in primitive residue classes, with constants depending only on fixed parameters.Source: §2 · p. 6

Generalized Elliott–Halberstam estimates

• The generalized Elliott–Halberstam conjecture GEHrϑs is stronger than EHrϑs and implies it for every fixed 0 < ϑ < 1.The paper formulates GEHrϑs through averaged discrepancy bounds for convolutions of sequences satisfying pointwise and Siegel–Walfisz-type conditions.Source: §2 · pp. 6–7
• GEHrϑs is known unconditionally for every fixed 0 < ϑ < 1/2 through the generalized Bombieri–Vinogradov theorem.The paper notes that this theorem is not directly used in its later analysis.Source: §2 · p. 7

Scope and limitations

• Extensions of MPZrϖ, δs to the generalized setting are unavailable except in restricted Type I/II or Type III configurations, limiting their usefulness here.The analysis therefore applies GEHrϑs effectively only when ϑ is very close to 1.Source: §2 · p. 7

Theorem 3.2

• Theorem 3.2 establishes DHL[50,2], DHL[35 410,3], DHL[1 649 821,4], DHL[75 845 707,5], and DHL[3 473 955 908,6] unconditionally, plus DHL[k,m+1] for k ≥ C exp((4 − 28/157)m).Under Elliott–Halberstam, it gives DHL[54,3], DHL[5511,4], DHL[41 588,5], DHL[309 661,6], and DHL[k,m+1] for k ≥ C exp(2m); under generalized Elliott–Halberstam, it gives DHL[3,2] and DHL[51,3].Source: §3 · p. 8

Theorem 3.3

• The cited passages give H(3) = 6, H(50) = 246, H(51) = 252, H(54) = 270, and H(5511) ≤ 52 116.Source: §3 · pp. 8–9

Sieve criterion

• Lemma 3.4 converts suitable weighted upper and lower asymptotics, together with a key inequality, into DHL[k,m+1] by forcing a positive weighted count with at least m+1 primes.The weights are optimized through the ratio (β1 + ··· + βk)/α.Source: §3 · pp. 9–10

Selberg sieve and asymptotics

• The sieve uses nonnegative squares of finite linear combinations of products of one-dimensional Selberg divisor sums, producing smooth weights concentrated where all n+h_i are almost prime.Theorem 3.5 and Theorem 3.6 supply prime and non-prime sum asymptotics under Elliott–Halberstam, Motohashi–Pintz–Zhang, or generalized Elliott–Halberstam hypotheses.Source: §3 · pp. 10–12

Distributional input

• The generalized Elliott–Halberstam non-prime asymptotic requires no support bound for one selected pair of functions, a flexibility crucial for DHL[3,2] and Theorem 1.4.The asymptotics provide limited independence, but no estimates involving two or more prime-weight factors, whose difficulty reaches that of the twin prime conjecture.Source: §3 · p. 13

Variational estimates

• Variational analysis on the standard simplex yields M54 > 4.00238, M5511 > 6, M41588 > 8, M309661 > 10, and M_k ≥ log k − C, implying parts (vii)–(xi).These multidimensional functions improve on earlier sieve restrictions to forms f(t1 + ··· + tk), which cannot make the analogous ratio exceed 4.Source: §3 · pp. 13–14

• The converse bound M_k ≤ k^(k−1) log k shows that the earlier bounds cannot be significantly improved, while Bombieri–Vinogradov gives a weaker DHL[r^54,2] consequence.Source: §3 · p. 14

Truncated-simplex sieve

• Sieving on a truncated simplex converts MPZ[ϖ,δ] input into DHL[k,m+1] whenever the associated truncated-simplex variational inequality holds.The support is restricted to t_1+⋯+t_k ≤ 1 within [0,α]^k.Source: §3 · p. 14

Applications of the truncated-simplex sieve

• The truncated-simplex variant supplies the parameter inequalities and numerical estimates used with Theorem 2.5 to establish parts (ii)–(vi) of Theorem 3.2, including the large-k asymptotic consequence in part (vi).For part (vi), the argument combines Theorems 3.11(vi), 2.5, and 3.10 to obtain DHL[k,m+1] for sufficiently large k under the stated parameter constraints.Source: §3 · p. 14

Epsilon-enlarged simplex

• The epsilon-enlarged-simplex theorem permits support on t_1+⋯+t_k ≤ 1+ε, trading the enlarged support for a reduced integration range and yielding DHL[k,m+1] under one of two hypotheses.Continuity of M_k,ε allows strict inequalities in parts (i) and (ii) to be replaced by non-strict ones; numerical computations give M_50,1/25 > 4.0043, M_4,0.168 > 2.00558, and M_51,1/50 > 4.00156.Source: §3 · p. 15

Computational limitations

• Other combinations of the sieve and prime-equidistribution theorems either failed to improve the numerology or were numerically infeasible to implement.The full-strength computation for Theorem 3.2(xii) is substantially more complicated than the hand-checkable computation for the weaker result.Source: §3 · pp. 15–16

Beyond epsilon enlargement

• Beyond epsilon enlargement, under GEH[ϑ], a test function supported on (k/(k−1))·R^k with vanishing marginal conditions can also imply DHL[k,m+1].The stated construction includes a symmetric piecewise-polynomial example for k=3 supported where t_1+t_2+t_3 ≤ 3, subject to the required vanishing marginal condition and inequality.Source: §3 · p. 16

4 Multidimensional Selberg sieves

• Lemma 4.1 supplies the key multidimensional sieve asymptotic for fixed smooth compactly supported functions, with extensions to modified denominator conditions.The proof uses Fourier expansions and rapid decay, factors the resulting expression as an Euler product, and invokes the simple pole of the Riemann zeta function at s = 1; related computations appear in.Source: §4 · pp. 16–19

4.1 The trivial case

• The trivial case follows by restricting to pairwise coprime divisor tuples, evaluating an arithmetic-progression sum, and showing the total error is O(x^(1−ε)).This argument proves case (i) of Theorem 3.6; a closely related estimate appears in [38, Lemma 6.2].Source: §4.1 · pp. 19–20

4.2 The Elliott-Halberstam case

• In the Elliott–Halberstam case, Chinese remainder concatenation produces primitive congruence conditions, while EH[ϑ] controls the resulting distributional error.The main term is obtained from Lemma 4.1, and replacing θ by Λ contributes an acceptable error of O(x^(1−(1−ϑ)/2)).Source: §4.2 · pp. 21–22

4.3 The Motohashi-Pintz-Zhang case

• The Motohashi–Pintz–Zhang case repeats the Elliott–Halberstam argument but derives the required estimate using squarefree moduli whose prime factors are bounded by x^δ.The hypotheses imply the relevant moduli are at most x^δ, allowing the notation and distributional framework of Claim 2.4 to be applied.Source: §4.3 · p. 22
• The Motohashi–Pintz–Zhang case follows by using uniform residue-class distribution from the Chinese remainder theorem and repeating the Section 4.2 argument under MPZ[ϖ, δ].Source: §4.3 · p. 22

4.4 Crude estimates on divisor sums

• Proposition 4.2 shows that the divisor sums λ_F are concentrated on almost primes, with related variants controlling a specified prime factor and least-prime-factor weights.The proof bounds alternating divisor sums using nonnegative expressions involving prime factors and standard combinatorial estimates; the exponent 1/10^k is sufficient but not optimal.Source: §4.4 · pp. 22–23
• Under Lemma 3.4’s hypotheses, Proposition 4.2 implies that ≫ x/log^k x integers n in [x,2x] have all n+h_i free of prime factors below x^ε, with at least m shifts prime.Source: §4.4 · p. 26

4.5 The generalized Elliott-Halberstam case

• In the generalized Elliott–Halberstam case, the proof treats λ_Fkλ_Gk as a substitute for θ, restricts n+h_i to almost primes using Proposition 4.2, and reduces the argument to one main-term and two error-term estimates.Source: §4.5 · pp. 26–27
• The GEH hypothesis yields the needed discrepancy bounds by factoring integers into large prime factors, partitioning them into short intervals, and expressing the resulting sets as convolutions.The method uses the fundamental theorem of arithmetic and the Siegel–Walfisz theorem [58, Satz 4; 34, Th. 5.29] to apply GEH after decomposing the indicator functions.Source: §4.5 · pp. 28–29
• The same factorization argument controls the remaining term, while repeated-prime contributions are negligible and square-free contributions admit a prime-number-theorem asymptotic.Repeated prime factors contribute O(x^(1−ε)), and the square-free sum is evaluated through logarithmic prime variables on the hyperplane t_1+...+t_r=1.Source: §4.5 · pp. 29–30
• The resulting multidimensional integrals converge absolutely, and iterating the integral identity shows the remainder L_a,k(F) tends to zero as k tends to infinity.This establishes the limiting identity used to complete the asymptotic analysis in the generalized Elliott–Halimberstam case.Source: §4.5 · pp. 30, 32

5 Reduction to a variational problem

• The common reduction res scales and shifts the function away from simplex boundaries, smooths it by convolution, and approximates it by finite sums of smooth tensor products.The regularization is designed to remain compatible with the vanishing marginal condition needed later.Source: §5 · p. 32, §5.1 · p. 33

5.1 Proof of Theorem 3.8

• Theorem 3.8 follows by constructing a nonzero smooth tensor-product approximation and applying the sieve estimates to obtain DHL[k, m+1].The approximation preserves the relevant integral and marginal quantities, while the resulting nonnegative sieve weight factorizes into one-dimensional components.Source: §5.1 · pp. 34–36

5.2 Proof of Theorem 3.10

• Theorem 3.10 uses the same tensor-product construction on a truncated simplex, replacing Theorem 3.5(i) and EH[ϑ] with Theorem 3.5(ii) and MPZ[ϖ, δ].The resulting components satisfy the support and sum conditions required to repeat the sieve argument.Source: §5.2 · p. 36

5.3 Proof of Theorem 3.12

• For Theorem 3.12, partitioning tensor factors and using either Theorem 3.6(i) or (ii) controls pairwise interactions, while a lower-bound argument avoids the epsilon loss preventing an asymptotic.The two cases are distinguished by the theorem’s hypotheses; in the second, the pigeonhole principle supplies an index to which Theorem 3.6(ii) applies.Source: §5.3 · pp. 37–38
• Theorem 3.12 follows by applying Theorem 3.5(i) under EHrϑs to a lower-bound construction whose component supports overlap only on a set of measure O(δ^3).Choosing δ sufficiently small and using Lemma 3.4 with (86) gives DHLrk, m ` 1s.Source: §5.3 · pp. 39–40

5.4 Proof of Theorem 3.14

• For Theorem 3.14, rescaling, translation, and regularization produce a smooth nonzero function preserving the vanishing marginal condition on the required region.The construction starts from F1, obtains F3 supported in a constrained region, and yields f3 with corresponding vanishing near boundary coordinates.Source: §5.4 · p. 40
• Theorem 3.14 is then reduced via Stone-Weierstrass to a finite sum of localized product functions satisfying the support inequalities needed for the sieve estimates.Each factor is smooth and supported on an interval of length at most δ^3, while the resulting components obey the displayed sum constraints.Source: §5.4 · p. 41
• Under GEHrϑs and EHrϑs, the resulting estimates combined with Lemma 3.4 prove DHLrk, m ` 1s when δ^3 is sufficiently small.The GEH hypothesis supplies the relevant estimate, while EH is invoked for the complementary bound.Source: §5.4 · pp. 41–42

Exact evaluation

• M_2 = 1/(1 − W(1/e)) = 1.38593... by locating a positive eigenfunction, where W is the Lambert W-function.The eigenfunction criterion identifies the corresponding eigenvalue with M_2.Source: §6 · p. 43

General upper bounds

• M_k ≤ k/(k − 1) log k for every k ≥ 2, although positive eigenfunctions are only conjectured to exist beyond k = 2.For k = 2 this gives M_2 ≤ 1.38629..., while the k = 3 bound 1.8454... shows that Theorem 3.8 alone cannot establish DHL[4,2] or DHL[3,2] even under GEH.Source: §6 · pp. 43–44

Perturbed quantities

• M_{k,ε} ≤ k/(k − 1) log(2k − 1) for k ≥ 2 and 0 ≤ ε < 1, so the constant improvement over M_k is asymptotically modest.For 1/3 ≤ ε < 1, the exact M_{2,ε} calculation shows that M_{2,ε} increases to 2 as ε approaches 1.Source: §6 · pp. 44–46

Lower bounds

• The restricted ansatz F(t_1,...,t_k)=f(t_1+...+t_k) gives M_k ≥ 4k/(k−1)j_{k−2}^2, including M_2 ≥ 1.383... and M_6 > 2.The M_6 bound recovers the Goldston–Pintz–Yıldırım result DHL[6,2], hence H_1 ≤ 16 under the Elliott–Halberstam conjecture, but this lower bound cannot force M_k above 4.Source: §6 · p. 46

Explicit lower-bound estimate

• Theorem 6.7 introduces an explicit parameterized estimate designed to remove the log log k term from the earlier lower bound M_k ≥ log k − 2 log log k − 2.The theorem defines an explicit function g and computable quantities Z, Z', W, X, V, and U under stated parameter constraints.Source: §6 · pp. 46–47

• The bound (107) also holds with M[T] when k ≤ M_k.Source: §6 · p. 47
• The proof converts the sieve integrals into probabilities involving independent variables X_1,...,X_k and partial sums S_k, yielding a bound in terms of P(S_k ≥ r).The variables are supported on [0,T] with a distribution determined by g(t)^2, and symmetry identifies the corresponding J_i(F) quantities.Source: §6 · pp. 47–48
• A random-function Cauchy–Schwarz argument bounds the main quantity using Y_1(r) and Y_2(r), after averaging over r from 1 to 1+τ.The resulting estimates reduce to controlling Z, Z_3, and the terms W X and V U.Source: §6 · pp. 48–49
• The Y_1 error term is controlled by logarithmic inequalities, symmetry, and the mean and variance of S_k, while the minor term Z_2 uses a crude quadratic bound.Each X_i has mean μ and variance σ^2, so S_k has mean kμ and variance kσ^2.Source: §6 · pp. 49–51
• The Y_2 term exploits the specific form of g through an identity for g(t)−h(t), then uses averaging in u and a split at 1+uτ−S_{k−1}=2c to handle a singular denominator.The near-boundary contribution is bounded by W X after a change of variables.Source: §6 · pp. 51–52

Possible refinement

• A small improvement to the bounds could be obtained by replacing the threshold 2c with an optimized parameter.This is noted as a possible refinement rather than part of the established bounds.Source: §6 · p. 52

Asymptotic parameter analysis

• The analysis establishes Z_3 = O(1), X = O(1), V = O(1), and U = O(1), and controls the remaining integral via a change of variables.Source: §6 · p. 54
• For sufficiently large k, hypotheses (104), (105), and (106) are obeyed; for C large enough, the resulting estimates imply Theorem 3.11(vi), with all implied constants effective.Source: §6 · p. 54

Remaining theorem cases

• Table 2 supplies parameter choices that yield the bounds for Theorems 3.9(viii)–(x) and Theorem 3.11(ii)–(v).For Theorem 3.11(ii)–(v), the choices correspond respectively to m = 2, 3, 4, 5, with ϖ and δ selected to satisfy the required conditions.Source: §6 · pp. 54–55, Table 2 · p. 55

7.1 Bounding M_k for medium k

• The authors formalize an observation of Maynard showing that, without loss of generality, the problem of lower bounding Mk may be restricted to symmetric functions.Source: §7.1 · p. 55

7.1 Bounding Mk for medium k

• A unit-cube-supported cutoff with ε = 0.18 also establishes Theorem 3.13(xii1) and simplifies the GEH-based proof of DHL(4,2).The simplification permits adding the hypothesis S(F_i0) + S(G_i0) < 1 to Theorem 3.6(ii).Source: §7.1 · p. 61
• M_4,ε ≥ 2.05411 and M_3,ε ≥ 1.91726 are obtained using optimized cubic-polynomial cutoffs.The M_4,ε bound uses ε = 5/21, while the M_3,ε bound uses ε = 56/113.Source: §7.1 · p. 61

7.4 Three-dimensional cutoffs

• The three-dimensional construction partitions the simplex into 60 symmetric polytopes and assigns degree-at-most-3 polynomials, reducing the cutoff search to a finite-dimensional quadratic program with linear marginal constraints.The partition combines coordinate permutations, epsilon splitting, and further subdivision needed to preserve graph structure.Source: §7.4 · pp. 62–63
• For ε = 1/4, the selected piecewise-polynomial cutoff obeys the six marginal conditions after numerical experimentation, quadratic-program optimization, rounding, and denominator clearing.The construction uses degrees 1, 2, and 3 on different polytope components and vanishes on D_xyz.Source: §7.4 · pp. 67–68

• The parity barrier prohibits sieve-theoretic methods from improving H_1 ≤ 6, even under the generalized Elliott-Halberstam conjecture.The discussion is explicitly informal and heuristic.Source: §8 · p. 68
• The H_1 ≤ 6 argument uses a non-negative weight supported on [x,2x], derived from averaged discrepancy bounds and asymptotic main-term control.The required estimates cover shifts h ∈ {0,2,6}, residue classes, and arithmetic functions including 1, Λ, and suitable Dirichlet convolutions.Source: §8 · p. 68
• Because the sieve argument remains valid after inserting an additional non-negative weight, comparable weighted discrepancy and main-term estimates would transfer any sieve-theoretic H_1 ≤ 4 proof to a stronger lower-bound setting.This invariance is the mechanism behind the parity obstruction.Source: §8 · pp. 68–69
• A Liouville-based weight vanishes on integers where at least two of n, n+2, and n+6 are prime, while predicted cancellation suggests the weighted estimates should still hold.The weight is built from pairwise Liouville correlations and is applied to the set of admissible prime patterns.Source: §8 · pp. 69–70
• The resulting contradiction blocks sieve-theoretic proofs of H_1 ≤ 4 and, by a simpler weight, the twin prime conjecture H_1 = 2.The same reasoning extends to related bounded-gap prime sets and rules out non-trivial sieve-theoretic lower bounds in those settings.Source: §8 · p. 70
• Circumventing the barrier would require stronger bilinear sieve axioms, but the paper reports no plausible way to control the needed bilinear expressions.Such axioms would likely imply non-trivial bounds leading toward the twin prime conjecture, whereas existing linear axioms do not.Source: §8 · pp. 70–71

Additional consequences

• Under GEH, sufficiently large multiples of 6 admit an n such that at least two of n, n−2, and x−n are prime.The same holds with n+2 replacing n−2.Source: §9 · p. 71
• This three-prime alternative implies either twin primes occur or one of x and x−2 is a sum of two primes.Source: §9 · p. 71

Proof strategy

• The construction uses a nonnegative sieve weight, asymptotic upper and lower bounds, the singular series, and a function supplied by Theorem 3.15.Source: §9 · pp. 71–72

Limitations

• Further numerical and analytic gains remain possible through improved variational bounds, distribution estimates, sieve regions, or non-Selberg constructions.The authors report that tested non-Selberg variants did not numerically improve the Selberg sieve, which appears locally optimal after cutoff optimization.Source: §9 · pp. 72–73

10 Narrow admissible tuples

• 246, 252, and 270 are the exact values H(50)=246, H(51)=252, and H(54)=270, with realizing tuples shown in Figures 1–3.These values follow from Clark and Jarvis.Source: §10 · p. 74, Figure 1 · p. 74, Figure 2 · p. 74

10.2.1 Fast admissibility testing

• Admissibility only needs testing modulo primes p<k, because a k-tuple cannot occupy every residue class modulo any prime p>k.Source: §10.2.1 · p. 74
• A boolean-vector test checks only selected residue ranges for most large primes, falling back to full enumeration when necessary and heuristically reducing complexity by O(log k).Source: §10.2.1 · p. 75

10.2.2 Sieving methods

• The sieving methods progress from Eratosthenes and Hensley–Richards to shifted Schinzel and shifted greedy variants, with the latter choices optimizing residue classes for larger primes.Source: §10.2.2 · pp. 75–76
• The Hensley–Richards construction gives H(k) ≤ k log k + k log log k − (1+log 2)k + o(k), while the basic Eratosthenes construction gives H(k) ≤ k log k + k log log k − k + o(k).Source: §10.2.2 · p. 76
• Local optimizations add little beyond better sieving algorithms, especially for large k, and the authors conjecture ⌈k log k+k⌉ as an eventual upper bound.Source: §10.2.2 · pp. 76–77

10.3.1 Improved time complexity

• For very large k, combining parameter optimization with admissibility testing is expected to reduce construction time from quasi-cubic to quasi-quadratic.Source: §10.3.1 · p. 77

10.3.2 Improved space complexity

• Gap encoding reduces storage from approximately k log k bits to O(k log log k) bits, while batched residue-class choices permit parallelized greedy sieving.Source: §10.3.2 · pp. 77–78, §10.3.3 · p. 78