DYNAMICS ON THE NUMBER OF PRIME DIVISORS FOR ADDITIVE ARITHMETIC SEMIGROUPS

The paper’s main statement (Theorem 1.3) matches the candidate solution’s goal, but the written proof has two gaps: (i) it defines μ_n via ∫f dμ_n := E_g∈G_n F((Ω(g)−log n)/√log n) f(T^{Ω(g)}x) and calls μ_n a probability measure, although its total mass is E_g F((Ω(g)−log n)/√log n)≠1 in general; and (ii) it does not explicitly justify the Gaussian normalization factor needed to identify the weak-* limit as (∫F dγ)·μ (i.e., the Erdős–Kac-type limit for Ω on G). The argument only shows T-invariance of limit points and hence convergence to c·μ for some c, but does not compute c. By contrast, the model solution supplies the missing probabilistic/analytic input (a two-variable Euler-product factorization, quasi-powers, CLT and LLT) and rigorously derives the Gaussian constant and the dynamical limit. These corrections are standard but necessary for completeness.