REVIEW
Clusters of primes revisited
Number Theory · Submitted 30 Oct 2026 · 16 references
Pith's verdict: New asymptotic lower bound for M_k that is worth citing; the H_m records are already superseded, and the main caveat is under-specified numerical certification, not the mathematics.
T0 · AI review
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Referee report
The paper
The paper claims
A product-of-reciprocals sieve function yields the strongest known bounds for gaps between primes in clusters
The reading
What carries the argument
The central object is the sieving function (3): $F(t_1,\dots,t_k) = Q(1+\varepsilon-P_1) \big/ \prod_{i=1}^k (1+at_i)$, where $Q$ is a polynomial and $a$ is a parameter taken as $a=\alpha k\log k$. The key machinery is the use of Laplace transforms to convert the $k$-fold convolution integrals $I_\varepsilon(F)$ and $J_\varepsilon(F)$ into single-variable contour integrals, combined with the explicit properties of the function $W(s;a) = \int_0^\infty e^{-st} (1+at)^{-2}\,dt$. This reduction makes the high-dimensional optimization tractable numerically and allows a closed-form asymptotic expansion for $M_k$. The paper also develops detailed estimates for the integrand, including monotonicity of $|W(\sigma+i\omega;a)|$ and tail bounds that enable rigorous certification of the numerical bounds.
Extended reading notes
Core claim
The paper claims that the function family $F(t_1,\dots,t_k) = Q(1+\varepsilon - P_1) / \prod_{i=1}^k (1+at_i)$, with $P_1 = \sum t_i$, $a \approx k\log k$, and a polynomial $Q$ of degree up to 20, provides a highly efficient basis for the Maynard–Tao sieve. Using the convolution structure of the key integrals $I_\varepsilon(F)$ and $J_\varepsilon(F)$, the author reduces the $k$-dimensional optimization to a single-variable Laplace transform, then computes the resulting Rayleigh quotient numerically with rigorous error bounds. The central numerical result is that $M_{k,\varepsilon} > 4m$ is achieved for $k$ as small as 3438 when $m=2$, leading to the claimed $H_m$ bounds. In the asymptotic regime, the paper explicitly evaluates the leading terms of $M_k$ and obtains a constant $c = 0.273572$ that measures the gap to the Polymath8b upper bound.
Load-bearing premise
The paper assumes that the specific family of sieving functions (3), with $Q$ a polynomial of degree at most 20 and $a=\alpha k\log k$, can approximate the true supremum of $M_k$ sufficiently well to yield the claimed numerical bounds and asymptotic constant, and that the numerical integration and error bounds implemented are correct and converge.
Editorial extensions
If this is right
- The bound $H_2 \leq 31\,140$ is a reduction by more than a factor of 12 compared to the previous best unconditional bound of $398\,130$, demonstrating that significant progress on prime cluster gaps is still possible by optimizing the sieve function alone.
- The asymptotic result $M_k \geq \log k - 0.273572 + o(1)$ shows that the product-of-reciprocals family already performs within 0.274 of the (non-optimal) Polymath8b upper bound, providing a benchmark for future sieve constructions.
- The explicit numerical bounds for $H_2$ through $H_5$ can be directly cited as the current record unconditional values until further improvements appear.
- The method of reducing $k$-dimensional Rayleigh quotients to one-dimensional integrals via Laplace transforms can be adapted to other Selberg-sieve optimization problems beyond prime gaps.
Reading between the lines
- The near-optimality of the family (3) suggests that the true supremum of $M_k$ may be exactly $\log k - c$ with $c$ very close to $0.273572$; confirming this would require proving an upper bound with the same constant.
- Combining the improved sieve function with stronger distributional hypotheses (such as the generalized Elliott–Halberstam conjecture) could push the $H_1$ bound below 6, as already shown in Polymath8b, but the present method may make that conditional bound easier to achieve with smaller $k$.
- The polynomial correction $Q$ provides diminishing returns as its degree increases (Table 3); a more general symmetric polynomial in the numerator of $F$ might yield further gains for finite $k$.
- The computational framework developed here (rigorous quadrature with tail bounds) could be applied to certify other high-dimensional variational problems arising in analytic number theory.
Original abstract
We revisit the generalized Rayleigh quotient optimization problem related to the Maynard-Tao sieve with the goal of finding a more efficient function basis. We demonstrate that a family based on products of reciprocals, previously used in asymptotic estimates for $H_m = \liminf_{n \to \infty}(p_{n+m}-p_n)$, provides results that are close to the theoretical limits of the sieve across many regimes. In particular, we establish $H_2 \leq 31 \, 140$, $H_3 \leq 2 \, 560 \, 022$, $H_4 \leq 193 \, 672 \, 056$, and $H_5 \leq 13 \, 659 \, 045 \, 840$. We also show that, as $k \to \infty$, the related variational quantity $M_k$, introduced by Maynard, satisfies the lower bound $M_k \geq \log k - 0.273572 + o(1)$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We revisit the generalized Rayleigh quotient optimization problem related to the Maynard-Tao sieve with the goal of finding a more efficient function basis. We demonstrate that a family based on products of reciprocals, previously used in asymptotic estimates for $H_m = \liminf_{n \to \infty}(p_{n+m}-p_n)$, provides results that are close to the theoretical limits of the sieve across many regimes. In particular, we establish $H_2 \leq 31 \, 140$, $H_3 \leq 2 \, 560 \, 022$, $H_4 \leq 193 \, 672 \, 056$, and $H_5 \leq 13 \, 659 \, 045 \, 840$. We also show that, as $k \to \infty$, the related variational quantity $M_k$, introduced by Maynard, satisfies the lower bound $M_k \geq \log k - 0.273572 + o(1)$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We revisit the generalized Rayleigh quotient optimization problem related to the Maynard-Tao sieve with the goal of finding a more efficient function basis. | contribution 0.65 | 0.64 | 0.05 | asserted 0.76 |
| s1 | We demonstrate that a family based on products of reciprocals, previously used in asymptotic estimates for $H_m = \liminf_{n \to \infty}(p_{n+m}-p_n)$, provides results that are close to the theoretical limits of the sieve across many regimes. | result 0.57 | 0.83 | 0.29 | asserted 1.00 |
| s2 | In particular, we establish $H_2 \leq 31 \, 140$, $H_3 \leq 2 \, 560 \, 022$, $H_4 \leq 193 \, 672 \, 056$, and $H_5 \leq 13 \, 659 \, 045 \, 840$. | result 1.00 | 0.95 | 0.48 | asserted 1.00 |
| s3 | We also show that, as $k \to \infty$, the related variational quantity $M_k$, introduced by Maynard, satisfies the lower bound $M_k \geq \log k - 0.273572 + o(1)$. | result 0.99 | 0.92 | 0.16 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:2610.41113 by the authors.
References
Discussion
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