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Functional Limit Theorems for Random Least Common Multiples

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Pith's one-line read Random LCMs obey functional large deviations, moderate deviations, and a Strassen LIL whose rates come from geometric-mark entropy and a Gaussian RKHS.

desk verdict Solid completion of the fixed-θ random-LCM program: functional LDP/MDP/LIL with clean entropy and RKHS rates, proofs that hold up. read the letter →

arxiv 2607.08129 v1 pith:WDRGUQJB submitted 2026-07-09 math.PR math.NT

Shaochen Wang , Guangyu Yang , Wang Zhou This is my paper · ORCID
classification math.PRmath.NT MSC 60F1060F1711N3760G50
keywords leastcommonmultiplerandomsetofintegersfunctionallargedeviationsmoderatelawtheiteratedlogarithmgeometricmarksreproducingkernelHilbertspaceprimenumbertheorem
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

When each integer up to n is kept independently with fixed probability θ, the least common multiple Ln of the retained set grows like exp(cθ n). This paper upgrades that growth law to pathwise large-deviation, moderate-deviation, and iterated-logarithm statements for the whole process t ↦ log L⌊nt⌋. After discarding an O(√n) arithmetic remainder, the process reduces to a triangular array of independent geometric marks on large primes; the large-deviation rate is the contraction of relative entropy of those marks, while the moderate-deviation rate and the almost-sure LIL cluster set are the quadratic form and unit ball of the reproducing-kernel Hilbert space of the already-known Gaussian limit. The results therefore complete the fixed-θ picture by giving precise exponential and almost-sure fluctuation scales for a classical arithmetic functional of a random set.

What carries the argument

The von Mangoldt identity reduces log LCM to a sum of prime indicators; discarding primes ≤ √n leaves independent geometric marks Gn,p on the large primes, which convert (via the prime-number theorem) into an entropy-contracted path measure for large deviations and into the covariance kernel Cθ for moderate deviations and the LIL.

What would settle it

Compute the exact difference log Ln − Sn(1) for large n and check whether it stays O(√n); if the ratio of that difference to the LDP, MDP or LIL scale fails to tend to zero, the reduction that underlies all three theorems collapses.

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Extended reading notes

Core claim

The polygonal interpolations of n⁻¹ log L⌊nt⌋ satisfy a large-deviation principle in C[0,1] with speed n/log n and good rate equal to the entropy contraction of geometric marks; after centering and moderate scaling they satisfy an MDP whose rate is half the squared RKHS norm of the Gaussian covariance of the CLT; and under the LIL normalization their almost-sure cluster set is exactly the unit ball of that RKHS.

Load-bearing premise

The claim that all primes up to the square root of n and all higher prime powers contribute only an O(square-root-n) error that is negligible on every fluctuation scale used later.

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

Referee Report

0 major / 5 minor

Summary. The paper studies the process t → log L_⌊nt⌋ for a Bernoulli random subset A_n of {1,…,n} with fixed retention probability θ ∈ (0,1). After the von Mangoldt reduction to large primes and an independent geometric-mark representation, the authors prove three functional limit theorems: an LDP for the polygonal processes of n^{-1} log L_⌊nt⌋ in C[0,1] with speed n/log n and good rate I_θ given by the entropy contraction of geometric marks (Theorem 2.1); an MDP for the centered and moderately scaled processes with speed b_n^{2} and rate ½∥f∥_{H_θ}^{2} (Theorem 2.3); and a Strassen-type LIL whose almost-sure cluster set is the unit ball of the same RKHS (Theorem 2.5). Endpoint and continuous-linear-functional consequences are derived by contraction. The proofs follow a standard route: finite-dimensional Gärtner–Ellis limits, exponential equicontinuity (monotonicity for the LDP; a local maximal inequality for the MDP/LIL), and identification of the projective rates with the entropy and RKHS contractions.

Significance. The work completes the fixed-θ asymptotic picture for random LCMs beyond the existing LLN and functional CLT of Alsmeyer–Kabluchko–Marynych. The rate functions are derived from first principles (relative entropy of geometric marks; the covariance kernel of the already-established Gaussian limit) rather than fitted, and the three theorems sit cleanly on the same arithmetic reduction. The explicit endpoint formulas, the local quadratic expansion of the LDP rate, the variational form of the MDP rate, and the numerical checks of finite-n Legendre transforms and importance-sampling tails are useful additions. The results are of genuine interest in probabilistic number theory and large-deviation theory for arithmetic functionals.

minor comments (5)
  1. The title of the manuscript and the running head contain a typographical space in “MUL TIPLES” / “COMMON MUL TIPLES”; this should be corrected before publication.
  2. Section 3 (Numerical simulations) is helpful, but the figure captions and Table 1 would be clearer if the precise values of n, θ, bin half-width, and number of tilted replications were repeated in every caption rather than only in the surrounding text.
  3. Remark 2.7 points to a companion working paper [24] on varying θ_n. A one-sentence pointer to the precise regimes (sparse / nearly complete) already treated there would help the reader place the fixed-θ results.
  4. In Appendix B the dual Euler equation is presented as supplementary. A brief remark that it is not used in the main proofs (already stated) could be moved into the introduction of the appendix so that readers who skip the appendix do not wonder whether the main rate identification depends on it.
  5. A few minor notational inconsistencies appear: m(x) is defined both as ⌊1/x⌋ and used as a running index; the same letter appears for the number of grid cells in the equicontinuity arguments. Distinct letters would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: rate functions are derived from geometric-mark entropy and the CLT covariance kernel by standard large-deviation arguments, not by fitting or self-referential definition.

full rationale

The paper’s derivation chain is self-contained and non-circular. The LDP rate I_θ is defined as the relative-entropy contraction of geometric marks (Eqs. 2.4–2.6) and then identified with the projective Legendre transform obtained from independent prime-coordinate log-mgfs via Gärtner–Ellis (Section 7); the definition is the candidate rate, not a quantity fitted from the process it is meant to govern. The MDP rate and LIL cluster set are the standard RKHS quadratic form associated with the covariance kernel C_θ of the already-established functional CLT; the paper re-derives the needed mean and covariance limits from the prime-number theorem (Lemmas 6.1–6.2) rather than importing them as black boxes. The only self-citations are to a companion working paper [24] on the varying-θ Poisson regime (Remark 2.7) and to an unrelated Engel-series note [23]; neither enters the fixed-θ proofs. The arithmetic reduction (Lemma 5.1) uses elementary Chebyshev bounds, not a self-proved uniqueness theorem. Numerical Section 3 validates endpoint formulas by importance sampling and finite-n cumulants; it does not fit parameters that later reappear as predictions. Consequently no step reduces by construction to its own input, and the circularity score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 2 invented entities

The paper works entirely inside classical probability and analytic number theory. The only free parameter is the model retention probability θ. Background axioms are standard (PNT, Chebyshev bounds, Gärtner–Ellis, relative-entropy calculus, RKHS theory). No new physical entities are postulated; the geometric marks and the entropy/RKHS rate functions are derived objects, not free inventions.

free parameters (1)
  • θ (Bernoulli retention probability) = any fixed value in (0,1); numerics use 1/2
    Fixed model parameter in (0,1); all rate functions and constants (c_θ, σ_θ^{2}, I_θ, J_θ) depend on it. Not fitted to data.
assumptions (6)
  • standard math Prime Number Theorem π(x)∼x/log x and Chebyshev bound ϑ(x)≤Cx
    Used throughout Sections 5–6 to convert prime sums into Riemann integrals and to bound the small-prime error by O(√n).
  • standard math Gärtner–Ellis theorem for finite-dimensional LDPs/MDPs
    Invoked in §§7.1 and 8.1 to pass from logarithmic moment generating functions to finite-dimensional rate functions.
  • standard math Relative entropy is lower-semicontinuous with compact sublevel sets; Fenchel–Moreau duality
    Used in §4 to identify the projective Legendre transform with the entropy contraction I_θ.
  • standard math Existence and basic properties of the reproducing-kernel Hilbert space of a continuous positive-definite kernel
    Used to identify the MDP rate and the LIL cluster set with the unit ball of H_θ (Proposition 4.6, Theorem 2.5).
  • standard math Talagrand/Arcones concentration for suprema of independent empirical processes and Freedman’s martingale inequality
    Appendix A supplies the local maximal estimate needed for exponential equicontinuity of the centered processes.
  • domain assumption Fixed retention probability θ∈(0,1) independent of n
    Stated in the introduction and Remark 2.7; the whole geometric-mark and Gaussian regime collapses if θ=θ_n→0 or 1.
invented entities (2)
  • Geometric marks G_{n,p} attached to large primes independent evidence
    purpose: Convert the independent large-prime indicators into a triangular array of weighted threshold processes whose entropy and covariance are explicit.
    Constructed on an enlarged probability space in §5; they are not free postulates but measurable functions of the original Bernoulli sequence plus independent tails.
  • Entropy contraction rate I_θ and RKHS rate J_θ independent evidence
    purpose: Serve as the good rate functions of the functional LDP and MDP/LIL respectively.
    Derived from relative entropy of geometric kernels and from the CLT covariance; not postulated ad hoc.

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Cite this review

Pith. "Pith review of Functional Limit Theorems for Random Least Common Multiples." pith.science (2026). https://pith.science/paper/WDRGUQJB

@misc{pith2026260708129,
  author       = {Pith},
  title        = {Pith review of: Functional Limit Theorems for Random Least Common Multiples},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDRGUQJB}},
  note         = {Machine review of arXiv:2607.08129}
}
abstract

Let $A_n$ be a subset of $\{1,2,\ldots,n\}$ obtained by retaining each integer independently with fixed probability $\theta\in(0,1)$, and let $L_n$ be the least common multiple of the integers in $A_n$. We prove a functional large deviation principle, a functional moderate deviation principle, and a Strassen-type functional law of the iterated logarithm for the process $(\log L_{\lfloor{nt}\rfloor})_{0\le t\le1}$. The large deviation rate function is given by an entropy contraction for geometric marks, while the moderate deviation rate function and LIL cluster set are described by the reproducing kernel Hilbert space associated with the Gaussian limit process.

Figures

Figures reproduced from arXiv: 2607.08129 by the authors.

Figure 1
Figure 1. shows the endpoint rate and its local MDP approximation. The quadratic approximation is accurate near the typical value and separates from the full LDP rate in the tails. 0.4 0.5 0.6 0.7 0.8 0.9 y 0.00 0.05 0.10 0.15 0.20 0.25 0.30 rate Endpoint LDP rate and local MDP approximation, = 1/2 I end (y) (y c )2 /(2 2 ) [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. displays this convergence. 0.5 0.6 0.7 0.8 0.9 y 0.00 0.05 0.10 0.15 0.20 0.25 0.30 rate Convergence of finite n large prime Legendre transforms limiting rate finite n rate, n = 5000 finite n rate, n = 20000 finite n rate, n = 100000 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Finite n endpoint LDP diagnostic for the large prime process. Squares show the first order saddlepoint bin approximation, and circles show IS estimates. Error bars are approximate 95% Monte Carlo intervals on the rate scale [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Centered finite n logarithmic moment generating functions and the limiting MDP quadratic cumulant for θ = 1/2. 4.1. Entropy and convex-dual representations of the LDP rate. Let K be the collection of measurable probability kernels x 7−→ νx = (νx(k))k≥1 from [0, 1] to N…

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