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Large prime gaps and probabilistic models

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Pith's one-line read A random residue-class sieve predicts that the largest prime gap below $x$ is governed by the interval-sieve inverse $g$ at the scale $2e^{-\gamma}(\log x)^2$.

desk verdict A genuinely new random-sieve model whose largest-gap theorem is rigorously reduced to the interval-sieve problem; the prime-gap prediction is a clearly labeled conjecture, not a theorem. read the letter →

arxiv 1908.08613 v3 pith:BYK7UBAC submitted 2019-08-22 math.NT math.PR

William Banks , Kevin Ford , Terence Tao This is my paper · ORCID
classification math.NTmath.PR MSC 11N0511B83
keywords primesprimegapsprobabilisticmodelsrandomsieveintervalHardy-Littlewoodconjecturesextremalboundslargest
topics The Collatz ConjectureThe Twin Prime Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes a new probabilistic model of the primes: an integer $n$ survives only if it avoids a randomly chosen residue class modulo each prime $p\le z(n)$, where the cutoff $z(n)\sim n^{1/e^\gamma}$ is tuned so that the survival density is about $1/\log n$. Working rigorously inside this model, the authors prove that the largest gap below $x$ is almost surely bracketed by $g((2e^{-\gamma}-o(1))(\log x)^2)$ and $g((2e^{-\gamma}+o(1))(\log x)^2)$, where $g$ inverts the extremal interval-sieve function $W_y\log y$. They conjecture the same bracket for the actual primes, which would make the maximal prime gap asymptotic to $g(2e^{-\gamma}(\log x)^2)$, and to $2e^{-\gamma}(\log x)^2$ under a folklore bound on the interval sieve. The model also satisfies the singular-series asymptotics to near-square-root error, and the paper proves a partial converse: any set obeying such uniform asymptotics must contain large gaps. The payoff is that the largest-gap problem is reduced to a deterministic extremal sieve question while the model still reproduces the expected local prime statistics.

What carries the argument

The load-bearing object is the random sieved set $R=\{n>e^2:n\in S_{z(n)}\}$, where $S_z$ is what remains after deleting one random residue class modulo each prime $p\le z$, and $z(t)$ is the largest prime with $\prod_{p\le z(t)}(1-1/p)^{-1}\le\log t$; the prime number theorem gives $z(t)\sim t^{1/e^\gamma}$. Large-gap statistics are controlled by the extremal interval-sieve quantity $W_y=\min_{\{a_p\}}|[0,y]\cap S_{(y/\log y)^{1/2}}|$, the minimum number of survivors of an interval of length $y$ after optimally choosing one deleted residue class at each prime up to $(y/\log y)^{1/2}$, and by its inverse $g(u)=\max\{y:W_y\log y\le u\}$. The proof transfers probability estimates for $|[0,y]\cap S_w|$ across five sieving ranges, using an upper-bound sieve, the large sieve with concentration inequalities, a martingale step, and a random-graph expansion for large primes.

What would settle it

For some explicit large $x$, find a prime-free interval of length greater than $g((\xi+\varepsilon)(\log x)^2)$, with $g$ computed from the extremal problem (1.11): Theorem 1.1 says the random model almost surely has no such gap, so such a prime gap would disprove Conjecture 1.2 in the upper direction; conversely, showing that for infinitely many $x$ no prime-free interval of length $g((\xi-\varepsilon)(\log x)^2)$ exists would disprove it in the lower direction.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a prime model built by deleting one uniformly random residue class for every prime $p\le z(n)$ admits a rigorous two-sided largest-gap law. For $g(u)=\max\{y:W_y\log y\le u\}$ and $\xi=2e^{-\gamma}$, one has almost surely $$g((\xi-o(1))(\log x)^2)\le G_R(x)\le g((\xi+o(1))(\log x)^2)$$ as $x\to\infty$. The same random set obeys the strong form of the singular-series conjecture for admissible tuples of size up to $\log^c x$ with near-square-root error, and the authors conjecture, as Conjecture 1.2, that the same gap bracket holds for the primes. They further prove deterministic conversions: uniform singular-series asymptotics for any set force large gaps, with averaged asymptotics forcing gaps above $g((c\xi-o(1))(\log x)^2)$, and they show that the existence of exceptional zeros would force $W_y$ infinitely often to be as small as $o(y/\log y)$, making the normalized largest gap unbounded.

Load-bearing premise

The load-bearing premise is that the primes' local structure is faithfully modeled by independent random residue classes up to the cutoff $z(n)\sim n^{1/e^\gamma}$; the rigorous results for the random model do not need this, but Conjecture 1.2 transfers them to the actual primes only if the premise holds.

Editorial extensions

If this is right

  • Almost surely, $G_R(x)$ lies between $g((\xi-o(1))(\log x)^2)$ and $g((\xi+o(1))(\log x)^2)$; with the current bounds on $g$, this ranges between $\xi(\log x)^2$ and $\xi(\log x)^2\log_2 x/(2\log_3 x)$.
  • Conjecture 1.2 transfers the same bracket to the primes, so $G_P(x)\sim g(\xi(\log x)^2)$; under the folklore $g(u)\sim u$, this becomes $G_P(x)\sim 2e^{-\gamma}(\log x)^2\approx1.1229(\log x)^2$.
  • Theorem 1.3 shows that the random model obeys the strong singular-series asymptotics with error $O(x^{1/2+o(1)})$ for admissible tuples of size up to $(\log x)^{1/2}$ in the stated range, so the model is consistent with both the Riemann Hypothesis analog and prime $k$-tuple statistics.
  • Any set satisfying the uniform singular-series asymptotic (1.17) must have gaps $\gg \kappa(\log x)^2/\log_2 x$, and the averaged version forces gaps above $g((c\xi-o(1))(\log x)^2)$.
  • If exceptional zeros exist, $W_y$ is infinitely often $o(y/\log y)$, and then $G_R(x)/(\log x)^2\to\infty$ almost surely; under Conjecture 1.2 the same unboundedness would hold for prime gaps.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper suggests a concrete numerical target: if the folklore $g(u)\sim u$ holds, $G_P(x)/(\log x)^2$ should tend to $2e^{-\gamma}\approx1.1229$; the $10^{18}$ record ratio is about $0.9206$, so the approach to the asymptotic may be slow and worth tracking computationally.
  • The same construction applies to admissible tuples: one can define an extremal survivor statistic $W_y^{(H)}$ for shifted tuples and expect a $g_H$ law for maximal gaps between twin primes or other constellations, a direction the paper lists as an open problem.
  • The connection to exceptional zeros means the sharp form of the prime-gap prediction cannot be separated from the exceptional-zeros problem: proving $G_P(x)=O((\log x)^2)$ under the model transfer would implicitly rule out the exceptional-zero scenario, and conversely a slow decay of exceptional zeros would push the largest gaps above any fixed multiple of $(\log x)^2$.
  • The deterministic converses imply that future work establishing the singular-series asymptotics in the required uniform range would immediately yield unconditional gap lower bounds of the same shape for the primes, with no probabilistic model needed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper introduces a new probabilistic model R of the primes: an integer n>e^2 belongs to R if it survives sieving by one uniformly random residue class modulo each prime p≤z(n), where z(n) is chosen so that Θ_{z(n)}^{-1} is approximately log n, i.e. z(n)∼n^{1/e^γ}. The central result, Theorem 1.1, states that with probability one the largest gap G_R(x) in R up to x satisfies g((ξ−o(1))log_2 x)≤G_R(x)≤g((ξ+o(1))log_2 x), where ξ=2e^{-γ} and g is the inverse of the extremal interval-sieve function W_y log y. The proof passes through five checkpoint sieving ranges (Lemmas 5.1–6.5), Borel–Cantelli arguments in Section 7, and a deterministic translation argument for the lower bound. The paper also proves a uniform Hardy–Littlewood type asymptotic for R (Theorem 1.3), a Riemann-hypothesis analogue (Theorem 1.4), deterministic large-gap consequences from Hardy–Littlewood type assumptions (Theorems 1.5 and 1.6), and a conditional theorem showing that exceptional zeros force W_y to be small infinitely often (Theorem 2.2). Conjecture 1.2 transfers the random-model gap asymptotic to the actual primes.

Significance. The paper is significant for several reasons. The model R is conceptually new: it is a sieve-based random set that automatically reproduces the Hardy–Littlewood singular series while remaining amenable to rigorous moment analysis, unlike the earlier Cramér and Granville models. Theorem 1.1 is a genuine almost-sure statement about a natural random sifted set, proved with substantial first- and second-moment estimates rather than heuristics. The formulation through the deterministic quantity W_y makes the connection between prime gaps, Jacobsthal's function, and the interval sieve problem precise, and it identifies the exact scale ξ log_2 x predicted by Granville's heuristic. The paper is also admirably honest: Conjecture 1.2 is explicitly labeled a conjecture, the dependence on the unknown behavior of W_y is encoded in the function g, and the exceptional-zeros connection is stated conditionally. If the transfer conjecture is correct, the paper yields the prediction G_P(x)∼g(ξ log_2 x), refining Cramér's model and matching Granville's lower-bound heuristic.

minor comments (4)
  1. [Abstract and §1.3] The abstract says a random residue class is selected for every prime modulus below a 'specific bound', but in (1.10) the cutoff z(n) depends on n. The wording should say 'a bound depending on n' to avoid confusion.
  2. [Theorem 7.1] In the displayed statement of Theorem 7.1 the expression g((1+ε)ξ(log x 2)^2) appears to be a typesetting artifact; it should presumably read g((1+ε)ξ(log_2 x)^2). Please correct this in the final version.
  3. [§1.6 table and §2.7] The table at the end of Section 1.6 lists the same asymptotic for Granville's model G as for R, but Section 2.7 gives only a short proof sketch for G. A footnote in the table stating that the G row is supported by a sketch rather than a full theorem would help readers distinguish rigorous results from heuristic ones.
  4. [Proof of Theorem 1.4] In the proof of Theorem 1.4 the parameter m is used both for the dyadic decomposition and for the exponent in x=2^m; renaming one of the two indices would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.1 is derived from the independent interval-sieve quantity W_y plus probabilistic estimates, and Conjecture 1.2 is a clearly labeled heuristic transfer.

full rationale

The paper's central rigorous result, Theorem 1.1, concerns the random set R, not the primes. R is defined in (1.10) purely through random residue classes and a density-matching cutoff z(t) chosen in (1.7). The largest-gap statement is then proved from the deterministic extremal interval-sieve quantity W_y, the definition g(u)=max{y : W_y log y <= u} in (1.13), and the probabilistic concentration estimates of Sections 5 and 6. No parameter in this derivation is fitted to prime-gap data or to G_R(x) itself; the cited bound (1.12) is standard Iwaniec linear-sieve input, and an improved W_y would refine, not change, the form of the theorem. The Hardy-Littlewood property of R in Theorem 1.3 is indeed built into the model through (1.8), but it is presented as a consistency property of the model, and it is not the mechanism that forces the gap bounds of Theorem 1.1. Conjecture 1.2 is explicitly labeled a conjecture transferring the same g to the primes; this is a recognized heuristic limitation rather than an unstated input or a fitted prediction. The same-author citation [11] is only background for the existing unconditional prime-gap lower bound and is not load-bearing for the derivation; no uniqueness theorem or ansatz is imported from the authors' prior work to forbid alternatives. The open problem in Section 1.7 about g(a)~g(b) and the remarks in Section 2.4 about ineffective bounds are honest limitations, not circular steps. Thus the derivation chain is self-contained with respect to its external sieve input, and there is no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data. The cutoff z(t) is determined by the requirement Theta_{z(t)}^{-1} <= log t, which matches the prime density via Mertens' theorem, and the constant 2e^{-gamma} is derived rather than tuned. Technical parameters in the proofs, such as v, w, U, and epsilon, are chosen for the argument and do not encode fitted values. The random set R is a mathematical construction, not a new physical entity, so the invented-entities list is empty.

assumptions (5)
  • standard math Prime number theorem and Mertens' theorem give Theta_z^{-1} = e^gamma log z + O(1), hence z(t) ~ t^{1/e^gamma}.
    Used in equation (1.7) to define the cutoff z(t) for the model R; this is standard analytic number theory.
  • standard math Known extremal sieve bounds on W_y from (1.12): 4y log_2 y / log^2 y is much less than W_y and W_y is at most y/log y plus an error term.
    Cited from Iwaniec's linear sieve and the trivial residue choice; these bounds set the scale of g and propagate into Theorems 1.1 and 1.5.
  • standard math Standard sieve and probability inequalities: upper bound sieve (Lemma 3.1), Azuma's inequality, Bennett's inequality, and Montgomery's large sieve.
    Invoked repeatedly in Sections 5 through 7 as accepted background tools.
  • domain assumption Conjecture 1.2: the actual primes P have gap function G_P(x) bounded below and above by g((xi plus or minus o(1)) log_2 x).
    This is the model-to-primes transfer; the paper labels it explicitly as a conjecture and does not claim a proof.
  • domain assumption Unproved regularities: g(a) ~ g(b) whenever a ~ b, and the folklore conjecture W_y ~ y/log y.
    Used only to state the simplified predictions G_P(x) ~ g(xi log_2 x) and G_P(x) ~ xi log_2 x; both are flagged as conjectural in Section 1.4 and Open Problem 2.

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Cite this review

Pith. "Pith review of Large prime gaps and probabilistic models." pith.science (2026). https://pith.science/paper/BYK7UBAC

@misc{pith2026190808613,
  author       = {Pith},
  title        = {Pith review of: Large prime gaps and probabilistic models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BYK7UBAC}},
  note         = {Machine review of arXiv:1908.08613}
}
abstract

We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval $[1,x]$. Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set.

Figures

Figures reproduced from arXiv: 1908.08613 by the authors.

Figure 1
Figure 1. GP(x) vs. various approximations asymptotics for prime k-tuples. Indeed, for any finite set H ⊂ Z, Cram´er’s model gives |{n 6 x : n + h ∈ C for all h ∈ H}| ∼ x log|H| x (x → ∞) with probability one, whereas the analogous assertion for prime numbers is false in general (for example, there is no integer n such that n + h is prime for all h ∈ {0, 1, 2}). The reason for the disparity is simple: for any prime p, every p… view at source ↗

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