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arxiv: 2605.23014 · v1 · pith:5AZRMXZDnew · submitted 2026-05-21 · 🧮 math.NT · math.PR

The Poisson Tail Conjecture for Primes in Short Intervals

Abhishek Jha This is my paper

Pith reviewed 2026-05-25 05:15 UTC · model grok-4.3

classification 🧮 math.NT math.PR
keywords primes in short intervalsPoisson distributionHardy-Littlewood conjectureprime gapssieve estimatesconcentration inequalitiesphase transition
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The pith

Conditional on strong Hardy-Littlewood conjectures, the number of primes in a random interval of length λ log x follows a Poisson distribution with mean λ when λ grows slowly.

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

Gallagher showed in 1976 that, under the Hardy-Littlewood conjectures, primes in a random short interval of fixed length λ log x are distributed as Poisson with mean λ. This paper extends the result to slowly growing λ, proving that the Poisson law still holds under a strengthened form of those conjectures. It also locates a phase transition beyond which the distribution deviates from Poisson and supplies explicit asymptotics for those deviations when λ grows slower than any fixed power of log x. The argument combines new extremal sieve bounds on short intervals with standard concentration tools.

Core claim

Conditional on a strong variant of the Hardy-Littlewood conjectures, the local counting function for primes in a randomly placed interval of length λ log x is asymptotically Poisson with parameter λ whenever λ tends to infinity slower than any positive power of log x. For larger λ still o((log x)^c) for every c, the paper derives the precise rate at which the distribution departs from Poisson, using extremal interval sieve estimates together with concentration inequalities.

What carries the argument

Extremal interval sieve estimates combined with concentration inequalities, applied to the strong Hardy-Littlewood conjectures to control the distribution of the prime counting function in short intervals.

If this is right

  • The normalized gaps between consecutive primes continue to follow an exponential distribution in the same slowly growing λ regime.
  • A phase transition occurs in the local statistics once λ exceeds the slowly growing range, with explicit deviation formulas supplied.
  • The Poisson tail conjecture holds rigorously throughout the range where λ grows slower than any fixed power of log x.
  • The same sieve-plus-concentration method yields control on higher moments and tail probabilities of the short-interval prime counts.

Where Pith is reading between the lines

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

  • If the strong conjectures hold, then random models for prime gaps remain valid over a wider range of interval lengths than previously established.
  • The identified phase transition supplies a concrete threshold at which deterministic arithmetic constraints begin to dominate random behavior.
  • Numerical sampling of short intervals for moderately large x could directly test the predicted deviation rates before the conjectures are settled.

Load-bearing premise

A strong variant of the Hardy-Littlewood conjectures supplies accurate enough asymptotics for primes in short intervals.

What would settle it

An explicit sequence of λ growing slower than any power of log x together with a concrete x at which the empirical distribution of prime counts in intervals of length λ log x deviates from Poisson(λ) by more than the error term allowed by the sieve bounds.

read the original abstract

In 1976, Gallagher showed that, conditional on the Hardy--Littlewood conjectures, the number of primes below $x$ in a randomly chosen short interval of length $\lambda \log x$ asymptotically follows a Poisson distribution with mean $\lambda$. Correspondingly, the normalized gaps between consecutive primes follow an exponential distribution, provided that the scaling parameter $\lambda$ is fixed. We investigate the validity and limitations of the associated folklore Poisson Tail Conjecture as $\lambda$ is allowed to grow. For \edit{slowly growing} $\lambda$, and conditional on a strong variant of the Hardy--Littlewood conjectures, we establish asymptotics demonstrating that the local counting statistics rigorously align with these predictions. Furthermore, we identify a phase transition and explore the breakdown of these distributions for larger $\lambda$, capturing the precise deviations when $\lambda$ grows slower than any fixed power of $\log x$. The proof relies on a novel combination of extremal interval sieve estimates and concentration inequalities from probability.

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 / 3 minor

Summary. The manuscript claims that, conditional on a strong variant of the Hardy--Littlewood conjectures, the number of primes in a random short interval of length λ log x follows a Poisson distribution with mean λ when λ grows slowly; it establishes the corresponding tail asymptotics via extremal interval sieves and concentration inequalities, identifies a phase transition, and gives precise deviations from the Poisson model when λ grows slower than any fixed power of log x but still tends to infinity.

Significance. Conditional on the stated conjectures, the results extend Gallagher's fixed-λ theorem to a growing range of λ and delineate the precise regime of validity of the Poisson model for short-interval prime counts. The combination of sieve estimates with probabilistic concentration tools is a methodological strength that yields explicit error control in the tails.

minor comments (3)
  1. [Abstract] The abstract and introduction should state the precise growth range for 'slowly growing' λ (e.g., λ = o(log log log x) or the explicit bound used in the theorems) rather than leaving it informal.
  2. [§1] The strong variant of the Hardy--Littlewood conjectures invoked in the main theorems should be written out explicitly (including the range of the moduli and the error term) in §1 or §2 so that the dependence is fully transparent.
  3. Notation for the normalized counting function and the Poisson parameter should be introduced once and used consistently; occasional switches between N(x;λ) and similar symbols hinder readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of the manuscript, as well as the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is conditional on external conjectures

full rationale

The paper derives Poisson tail asymptotics and phase transitions for slowly growing λ explicitly conditional on a strong variant of the Hardy-Littlewood conjectures (external to the paper). The proof combines extremal interval sieve estimates with concentration inequalities from probability theory; no load-bearing step reduces to a self-definition, fitted input renamed as prediction, or self-citation chain. The argument is self-contained against the stated external hypothesis and contains no internal reduction by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the Hardy-Littlewood conjectures as the primary external assumption; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Strong variant of the Hardy-Littlewood conjectures
    All stated asymptotics and the phase transition are conditional on this strengthened form of the conjectures.

pith-pipeline@v0.9.0 · 5695 in / 1192 out tokens · 23922 ms · 2026-05-25T05:15:34.009344+00:00 · methodology

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Reference graph

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