arXiv:2610.28328v1
A Joint Local Limit Theorem for Prime Divisor Counting Functions
math.NTmath.DSmath.PR11N6011N3711K6537A44
Abstract
We establish a uniform two-dimensional local limit theorem for the joint distribution of and . More precisely, along a set of scales of logarithmic density one, the normalized joint counts are uniformly approximated by the density of two independent Gaussian random variables, each with mean and variance , up to a power-saving error. The proof builds upon a recent circle-method framework of Tao and Teräväinen. Our results equally hold for the additive function . Additionally, we explore applications of our local limit theorem in the setting of uniquely ergodic systems.
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Detailed mathematical audit
01Statements2 reported findingsNo statement issues found
The joint local limit theorem for the two prime-divisor counting functions and its uniquely ergodic consequences are supported in the stated logarithmic-density sense.
Theorem 1.1No issue found
The bivariate local limit theorem has the claimed uniformity
PDF pp. 2–5 and 9–31 · arXiv:2610.28328v1
The high-frequency range is controlled by the Tao–Teräväinen input, while the sieve transfer couples the arithmetic functions to an independent prime model. The bivariate characteristic-function expansion and smoothing argument are uniform in the two lattice coordinates and in all four choices. The exceptional scales have logarithmic density zero as stated.
Theorem 1.2No issue found
The dynamical consequences follow from total-variation control
PDF pp. 5–7 and 31–34 · arXiv:2610.28328v1
The local limit theorem gives total-variation approximation by the product Gaussian lattice law on the relevant windows. Unique ergodicity controls the resulting averages uniformly in the base point, and the tails are negligible by the established moment bounds.
02Proofs2 reported findingsNo proof issues found
The dyadic reduction, sieve coupling, random-model local limit theorem, truncation, smoothing, and dynamical transfer arguments are correct and complete.
Sections 3–4No issue found in the reviewed proof
The arithmetic functions are transferred to one common random model
PDF pp. 9–26 · arXiv:2610.28328v1
The dyadic selection uses one common scale , and the fundamental-lemma error is uniform in both coordinates. Proposition 4.6 correctly couples the and variants, including repeated prime powers, before the two-dimensional characteristic-function estimate is applied.
Theorem 4.8 and Section 5No issue found in the reviewed proof
Fourier inversion and smoothing recover the full local statement
PDF pp. 26–34 · arXiv:2610.28328v1
The small-frequency Taylor expansion has the required covariance and uniform remainder, the medium and large frequencies are negligible, and truncation errors are summable on the retained scales. Fourier inversion therefore yields the claimed lattice probabilities, and the same uniform estimates justify the final dynamical averaging step.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.