Variance and Central Limit Theorem for the Log Coprime Density of Random LCMs

This paper studies the second-order probabilistic behaviour of the coprime density associated with the least common multiple of randomly sampled integers. The model considers independent random integers chosen uniformly from a finite range and examines how the coprime density of their least common multiple behaves when both the range size and the number of samples grow together. The paper proves an explicit asymptotic formula for the variance of the centred logarithmic coprime density and establishes a central limit theorem showing that its normalized fluctuations become Gaussian. It also derives a corresponding normal limit for the coprime density on its original scale. The variance analysis is based on a decomposition into diagonal and off-diagonal prime contributions. The diagonal term is evaluated through asymptotic estimates for primes grouped according to the number of their multiples in the sampling range. The off-diagonal term is shown to be negligible using an exact covariance identity for suitable prime pairs. The central limit theorem is proved through a martingale decomposition. The main technical ingredients are a uniform bound on the martingale increments and concentration of the conditional variance. These are obtained by combining coupon-collector estimates, sieve-based geometric observations, explicit moment bounds for prime hitting times, covariance estimates, level summation, and an L2L^2L2 Minkowski argument controlling the dependence among prime-pair terms. The deposit includes the complete manuscript, LaTeX source, compiled PDF, deterministic code for finite-range variance calculations, Monte Carlo simulations, raw numerical outputs, generated figures, automated tests, citation metadata, and documentation for reproducing the computational results. This work continues the first-order analysis developed in the companion paper, “On the Expected Coprime Density of the Least Common Multiple of Random Integers,” DOI: 10.5281/zenodo.21325880.