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hexagon:2610.00419v1

math.NT — Number Theory

Improved bounds on HmH_m for 2≤m≤52\le m\le 5

Contributed by Annanay Kapila

Let Hm=lim inf⁡n→∞(pn+m−pn)H_m=\liminf_{n\to\infty}(p_{n+m}-p_n), where pnp_n denotes the nnth prime. We prove \[ H_2\le 20{,}370,\qquad H_3\le 1{,}380{,}156,\qquad H_4\le 82{,}959{,}170,\qquad H_5\le 5{,}040{,}082{,}836, \] improving the bounds 33,11833{,}118, 2,718,1082{,}718{,}108, 214,099,720214{,}099{,}720 and 14,541,349,28814{,}541{,}349{,}288 of Roeling (2026). We use the multidimensional Selberg sieve of Maynard and Tao in the form developed by Polymath. The arithmetic input is Stadlmann's equidistribution estimate for primes in arithmetic progressions, in the extension to triply densely divisible moduli given in OpenAI's 2026 preprint proving H1≤186H_1\le186. Together with a criterion from the same preprint for the triple dense divisibility of least common multiples, it allows the sieve weights to be supported, instead of on Polymath's truncated simplex, on a region in which single coordinates may be as large as about 0.380.38. On this region we use product trial functions multiplied by an optimised function of the sum of the coordinates, and we certify the resulting variational quotients in ball arithmetic, computing the relevant distribution functions by Fourier inversion.

Primarily AI-generated textHuman understanding: some parts
Oct 20, 2026PDF ↗View work →
hexagon:2610.00380v1

math.NT — Number Theory

Note on variable role-reversals

Contributed by Runbo Li

In [1], the author uses variable role-reversals only when θ<4589−ε\theta < \frac{45}{89} - \varepsilon. In this note, we shall discuss how to use variable role-reversals when 4589−ε⩽θ<1120−ε\frac{45}{89} - \varepsilon \leqslant \theta < \frac{11}{20} - \varepsilon.

Primarily human-written textHuman understanding: all partsHarman's sieveVariable role-reversalsarithmetic progressionssieve methods
Oct 20, 2026PDF ↗View work →
hexagon:2610.00352v1

math.NT — Number Theory

Prime perturbations of polynomial observations: instability and incidence criteria

Contributed by Brice Pouly

The cost of changing prime signs depends on how the resulting multiplicative function is observed. On a fixed prime support of finite harmonic sum, we prove that existence of a polynomial mean for every sign assignment is equivalent to uniform prime truncation and uniform convergence to local products. The criterion applies separately to ordinary and logarithmic means. For independent activations with finite expected harmonic cost, the almost-sure truncation defect is the averaged large-prime parity probability, not a sum of incidences. These criteria separate averaging methods and observation geometries. One common support is logarithmically stable for every fixed polynomial and every assignment, yet a fixed perturbation of (−1)v2(n)(-1)^{v_2(n)} on that support has divergent ordinary means on n2+1,n2+2,n2−2n^2+1,n^2+2,n^2-2. Collective upper sieves give stability at critical counting exponents without critical summability, and harmonic stability on rectangles whose side lengths have asymptotically equal logarithms. For widths XαX^\alpha, Gaussian prime estimates give the matching universal budget threshold when 119/500<α<1119/500<\alpha<1; the negative result concerns only the sum of two squares. Around every completely multiplicative sign function, fixed perturbations of arbitrarily small ∑p−σ\sum p^{-\sigma} cost can have divergent logarithmic means on n2+1n^2+1 for 125/164<σ≤1125/164<\sigma\leq1. A positive oscillation bound is common to this range, although the starting scales may depend on σ\sigma. Window laws further distinguish all origins from almost all origins.

Primarily AI-generated textHuman understanding: no partsharmonic costmultiplicative functionsordinary and logarithmic meansparity activationspolynomial observationsprime perturbationssieve methods
Oct 20, 2026PDF ↗View work →

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