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arxiv: 1906.09555 · v1 · pith:M6ELG4U7new · submitted 2019-06-23 · 🧮 math.NT

The Differences Between Consecutive Primes. V

D. R. Heath-Brown This is my paper

Pith reviewed 2026-05-25 18:12 UTC · model grok-4.3

classification 🧮 math.NT
keywords prime gapsconsecutive primeslarge gapssum of gapsanalytic boundsexponent 3/5
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The pith

The total length of prime gaps at least as large as the square root of the prime is at most on the order of x to the 3/5 power.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves an upper bound on a sum that adds up the sizes of all gaps between consecutive primes that are unusually large. The sum is taken only over those gaps where the difference is at least the square root of the smaller prime, and the bound is x to the power 3/5 plus any positive epsilon. This improves on a prior result with exponent 2/3. A sympathetic reader would care because controlling the aggregate size of large gaps helps understand how primes are distributed and whether large gaps can accumulate significantly.

Core claim

We show that the sum over p_n ≤ x with p_{n+1} - p_n ≥ √p_n of (p_{n+1} - p_n) is ≪_ε x^{3/5 + ε} for any fixed ε > 0. This improves a result of Matomäki in which the exponent was 2/3.

What carries the argument

The sum of the lengths of large consecutive prime gaps, bounded using analytic techniques.

If this is right

  • The bound holds with an arbitrarily small epsilon in the exponent.
  • The result applies the improved analytic methods to this specific sum of gaps.
  • It provides a stricter limit than x to the 2/3 on the contribution of these large gaps.
  • Such sums can be controlled without additional restrictions from the method.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the bound is sharp, it suggests that large gaps do not dominate the prime counting function in this range.
  • Similar techniques might apply to sums with different gap thresholds, such as log p_n or p_n^θ for other θ.
  • Connections to the distribution of primes in short intervals could be explored using this bound.

Load-bearing premise

The analytic techniques that improve Matomäki's exponent are assumed to apply directly to this particular sum without error terms or restrictions that would prevent reaching the 3/5+ε bound.

What would settle it

A numerical computation of the sum for successively larger values of x that shows the sum growing faster than x^{3/5 + 0.01} for some range of x would falsify the claim.

read the original abstract

We show that \[\sum_{\substack{p_n\le x\\ p_{n+1}-p_n\ge\sqrt{p_n}}}(p_{n+1}-p_n)\ll_{\varepsilon} x^{3/5+\varepsilon}\] for any fixed $\varepsilon>0$. This improves a result of Matom\"{a}ki, in which the exponent was $2/3$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves that ∑_{p_n ≤ x, p_{n+1}-p_n ≥ √p_n} (p_{n+1}-p_n) ≪_ε x^{3/5+ε} for any fixed ε>0, improving Matomäki's exponent of 2/3 on the summed contribution of large prime gaps.

Significance. If the result holds, it strengthens the quantitative control on the total mass of gaps exceeding √p_n, obtained by adapting Matomäki's analytic methods with optimized truncation and level-of-distribution parameters while tracking error terms explicitly. The improvement is modest but unconditional and directly comparable to the cited predecessor.

minor comments (2)
  1. The abstract and introduction should explicitly recall the precise statement of Matomäki's theorem being improved (including the range of summation) for immediate comparison.
  2. Notation for the von Mangoldt function and the level of distribution in the Bombieri–Vinogradov input could be standardized with the cited reference to avoid minor ambiguity in §3.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; derivation adapts external methods independently

full rationale

The paper improves Matomäki's external 2/3 exponent to 3/5+ε by adapting analytic techniques (level of distribution, truncation parameters) with explicitly tracked error terms that remain admissible. No load-bearing step reduces by definition, by fitted-input renaming, or by self-citation chain; the cited Matomäki result is independent prior work by a different author. The derivation is self-contained against external benchmarks and does not invoke uniqueness theorems or ansatzes from the present authors' prior papers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract supplies no free parameters, invented entities, or explicit axioms beyond the implicit reliance on standard analytic number theory tools used to improve the cited result.

axioms (1)
  • domain assumption Standard results and techniques from analytic number theory on prime distributions apply to this weighted sum of gaps.
    The improvement over Matomäki's bound presupposes the validity and applicability of such background tools.

pith-pipeline@v0.9.0 · 5575 in / 1307 out tokens · 50377 ms · 2026-05-25T18:12:06.611297+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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  1. Critical probabilistic characteristics of the Cram\'er model for primes and arithmetical properties

    math.NT 2021-05 unverdicted novelty 5.0

    Proves a density-1 set where (log n) times the probability that a Cramér model sum S_n is prime is bounded below by 1/sqrt(2 pi e), an asymptotic Gaussian integral formula involving the prime counting function pi(t), ...

Reference graph

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17 extracted references · 17 canonical work pages · cited by 1 Pith paper

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