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arxiv: 2105.11020 · v1 · pith:ZVR7TX7Dnew · submitted 2021-05-23 · 🧮 math.NT

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

Michel Weber This is my paper

Pith reviewed 2026-05-24 13:35 UTC · model grok-4.3

classification 🧮 math.NT
keywords Cramér modelrandom primesBernoulli sumslocal limit theoremquasiprimesprime number theoremSturm-Liouville spectrum
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The pith

There exists a density-1 set of n such that the Cramér model's probability that S_n is prime is at least 1 over sqrt(2 pi e) times 1 over log n.

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

The paper studies the Cramér probabilistic model in which each integer j is retained independently with probability 1/log j to build the random sum S_n. It proves that along a set S of natural numbers with asymptotic density 1, the liminf of (log n) times the probability that S_n is prime is bounded below by 1 over sqrt(2 pi e). The same set also satisfies an explicit integral formula for that probability derived from the local central limit theorem applied to the mean m_n and variance B_n of S_n. Additional results give lower bounds on the probability of zeta-quasiprimes and show that the model implies the prime-number-theorem error term varies with the choice of subsequence.

Core claim

In the Cramér model with independent Bernoulli random variables xi_j of success probability 1/log j, letting S_n denote the partial sum, there exists a set S of density 1 such that liminf over n in S of (log n) P{S_n is prime} is at least 1/sqrt(2 pi e). Along the same set the probability admits the local-limit expression (1+o(1))/sqrt(2 pi B_n) times the integral of the Gaussian density against d pi(t) over an interval of width sqrt(2 b B_n log n) centered at m_n. The model also yields a positive lower bound (1-eta) e^{-gamma}/log zeta for the probability that a truncated sum S'_n is zeta-quasiprime when zeta grows slower than exp(c log n / log log n).

What carries the argument

The Cramér model: independent Bernoulli random variables xi_j with P(xi_j=1)=1/log j for j>=2, and S_n their partial sum up to n, whose distribution is analyzed via the central limit theorem and local limit theorems with mean m_n and variance B_n.

If this is right

  • The explicit local-limit integral formula for P{S_n prime} holds for all n in the density-1 set S.
  • For any eta>0 the probability that the truncated sum S'_n is zeta-quasiprime is at least (1-eta) e^{-gamma}/log zeta when zeta is not too large.
  • With probability 1 the model eventually avoids certain infinite sequences of actual primes.
  • The prime-number-theorem error term predicted by the model depends on the subsequence along which it is measured.
  • The lengths and frequencies of intervals on which the normalized deviation |S_n - m_n|/sqrt(B_n) stays bounded are governed by the spectrum of a Sturm-Liouville equation.

Where Pith is reading between the lines

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

  • The lower bound on primality probability along most n could be compared with the actual distribution of primes in short intervals or along arithmetic progressions.
  • The subsequence sensitivity of the error term suggests that any deterministic analogue would require controlling oscillations of the Chebyshev function along different sets of density 1.
  • The quasiprime lower bound might be tested numerically for moderate zeta to see how closely the model tracks known counts of smooth or almost-prime integers.

Load-bearing premise

The xi_j are independent Bernoulli random variables with success probability exactly 1/log j, which allows the central limit theorem and local limit theorems to be applied directly to S_n.

What would settle it

A direct computation or simulation of the actual liminf of (log n) P{S_n is prime} over all density-1 sets that yields a value strictly smaller than 1/sqrt(2 pi e).

read the original abstract

This work is a probabilistic study of the 'primes' of the Cram\'er model. We prove that there exists a set of integers $\mathcal S$ of density 1 such that \begin{equation}\liminf_{ \mathcal S\ni n\to\infty} (\log n)\mathbb{P} \{S_n\ \hbox{prime} \} \ge \frac{1}{\sqrt{2\pi e}\, }, \end{equation} and that for $b>\frac12$, the formula \begin{equation} \mathbb{P} \{S_n\ \text{prime}\, \} \, =\, \frac{ (1+ o( 1) )}{ \sqrt{2\pi B_n } } \int_{m_n-\sqrt{ 2bB_n\log n}}^{m_n+\sqrt{ 2bB_n\log n}} \, e^{-\frac{(t - m_n)^2}{ 2 B_n } }\, {\rm d}\pi(t), \end{equation} in which $m_n=\mathbb{E} S_n,B_n={\rm Var }\,S_n$, holds true for all $n\in \mathcal S$, $n\to \infty$. Further we prove that for any $0<\eta<1$, and all $n$ large enough and $ \zeta_0\le \zeta\le \exp\big\{ \frac{c\log n}{\log\log n}\big\}$, letting $S'_n= \sum_{j= 8}^n \xi_j$, \begin{eqnarray*} \mathbb{P}\big\{ S'_n\hbox{\ $\zeta$-quasiprime}\big\} \,\ge \, (1-\eta) \frac{ e^{-\gamma} }{ \log \zeta }, \end{eqnarray*} according to Pintz's terminology, where $c>0$ and $\gamma$ is Euler's constant. We also test which infinite sequences of primes are ultimately avoided by the 'primes' of the Cram\'er model, with probability 1. Moreover we show that the Cram\'er model has incidences on the Prime Number Theorem, since it predicts that the error term is sensitive to subsequences. We obtain sharp results on the length and the number of occurrences of intervals $I$ such as for some $z>0$, \begin{equation}\sup_{n\in I} \frac{|S_n-m_n|}{ \sqrt{B_n}}\le z, \end{equation} which are tied with the spectrum of the Sturm-Liouville equation.

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

3 major / 2 minor

Summary. The paper studies the Cramér probabilistic model with S_n = sum_{j=2}^n ξ_j where the ξ_j are independent Bernoulli r.v.s with P(ξ_j=1)=1/log j. It proves existence of a density-1 set S such that liminf_{n→∞, n∈S} (log n) P(S_n prime) ≥ 1/sqrt(2πe), derives an asymptotic formula for this probability as a Gaussian integral against dπ(t), establishes lower bounds on P(S'_n is ζ-quasiprime) for ζ up to exp(c log n / log log n), examines sequences of primes avoided by the model with probability 1, and obtains results on the length and frequency of intervals where |S_n - m_n| / sqrt(B_n) is bounded, linked to a Sturm-Liouville spectrum.

Significance. If substantiated, the results supply explicit quantitative heuristics from the Cramér model for prime probabilities along density-1 subsequences and for quasiprimes, together with a link between model fluctuations and the prime-number-theorem error term; the connection to the Sturm-Liouville equation for interval statistics is a distinctive technical feature.

major comments (3)
  1. [Abstract, Eq. (2)] Abstract, displayed formula (2): the probability P{S_n prime} is written as an integral of the local-limit Gaussian against dπ(t). Because π(t) is the actual prime-counting function, the model probability is expressed in terms of external prime data rather than being derived solely from the distribution of the ξ_j; this dependence is load-bearing for both the displayed asymptotic and the subsequent claim that the model predicts sensitivity of the PNT error term to subsequences.
  2. [Abstract, local-limit statement] Abstract and § on local-limit application: the existence of the density-1 set S and the validity of the integral formula rest on the local-limit theorem supplying P(S_n = k) = (1+o(1)) (2π B_n)^{-1/2} exp(-(k-m_n)^2/(2 B_n)) uniformly for all integers k inside an interval of width O(sqrt(B_n log n)) centered at m_n, for all n∈S. No explicit error bounds, Lyapunov-ratio estimates, or characteristic-function decay controls are supplied to justify that the o(1) remains valid on a density-1 set when the window reaches sqrt(log n) standard deviations; without such controls the central liminf claim cannot be verified.
  3. [Abstract, liminf claim and quasiprime result] The liminf inequality and the quasiprime lower bound both invoke the same local-limit approximation on S; if the uniformity fails, both quantitative statements are unsupported.
minor comments (2)
  1. [Notation] The summation range for the main sum S_n is never stated explicitly (contrast with the explicit range j=8 to n for S'_n).
  2. [References] The paper invokes “standard” local-limit theorems for non-i.i.d. lattice sums but supplies no reference list entry for the precise version used.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments. We address each major point below. The integral formula is intentional and we will clarify its motivation; we agree that additional details on the local-limit controls are needed and will supply them in revision.

read point-by-point responses

  1. Referee: [Abstract, Eq. (2)] Abstract, displayed formula (2): the probability P{S_n prime} is written as an integral of the local-limit Gaussian against dπ(t). Because π(t) is the actual prime-counting function, the model probability is expressed in terms of external prime data rather than being derived solely from the distribution of the ξ_j; this dependence is load-bearing for both the displayed asymptotic and the subsequent claim that the model predicts sensitivity of the PNT error term to subsequences.

    Authors: The integral arises directly from summing the local-limit approximation over prime values of k, which is equivalent to integrating the Gaussian density against dπ(t). This is deliberate: the Cramér model is employed precisely to generate heuristics that relate model probabilities to the actual locations of primes. The dependence on π(t) is the mechanism by which the model predicts that PNT error terms affect prime probabilities along different subsequences. We will add a short clarifying sentence in the abstract and introduction to make this motivation explicit. revision: partial

  2. Referee: [Abstract, local-limit statement] Abstract and § on local-limit application: the existence of the density-1 set S and the validity of the integral formula rest on the local-limit theorem supplying P(S_n = k) = (1+o(1)) (2π B_n)^{-1/2} exp(-(k-m_n)^2/(2 B_n)) uniformly for all integers k inside an interval of width O(sqrt(B_n log n)) centered at m_n, for all n∈S. No explicit error bounds, Lyapunov-ratio estimates, or characteristic-function decay controls are supplied to justify that the o(1) remains valid on a density-1 set when the window reaches sqrt(log n) standard deviations; without such controls the central liminf claim cannot be verified.

    Authors: We acknowledge that the current text does not display explicit Lyapunov ratios or characteristic-function estimates. The local-limit statement for the Poisson-binomial sums with p_j = 1/log j is invoked via a standard theorem for independent non-identical summands; the density-1 set S is obtained by excising a zero-density collection of n where the error term exceeds the required threshold. To meet the referee’s concern we will insert a dedicated paragraph (or short subsection) that records the precise local-limit theorem applied, states the Lyapunov condition satisfied by the variances B_n ∼ n/log n, and explains how the exceptional set is shown to have density zero. revision: yes

  3. Referee: [Abstract, liminf claim and quasiprime result] The liminf inequality and the quasiprime lower bound both invoke the same local-limit approximation on S; if the uniformity fails, both quantitative statements are unsupported.

    Authors: Both the liminf lower bound and the ζ-quasiprime estimate rely on the local-limit approximation holding uniformly on the same density-1 set S (the quasiprime argument uses the truncated sum S'_n but the same error-control mechanism). Once the additional local-limit controls requested above are supplied, the uniformity on S will be justified and both claims will be supported. We will also verify that the quasiprime range ζ ≤ exp(c log n / log log n) stays inside the window where the local-limit error remains controllable. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies standard LLT to model and couples with external π(t)

full rationale

The paper defines the Cramér model via independent Bernoulli ξ_j with p_j=1/log j, applies the local limit theorem to approximate the distribution of S_n, and rewrites P(S_n prime) as an integral of the Gaussian density against dπ(t). This is a direct, non-tautological rewriting of the sum over primes of the point masses; the resulting liminf lower bound is a theorem about the interaction between the model and the actual prime locations, not a reduction of the claim to its own inputs by construction. No self-citations, fitted parameters renamed as predictions, or self-definitional steps appear. The derivation is self-contained against external benchmarks (LLT, prime number theorem properties) and receives score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on the standard definition of the Cramér model together with classical limit theorems for sums of independent random variables.

axioms (1)
  • domain assumption The random variables xi_j (j>=2) are independent with P(xi_j=1)=1/log j
    This is the defining assumption of the Cramér model invoked for all statements about S_n and S'_n.

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Reference graph

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