On the Expected Coprime Density of the Least Common Multiple of Random Integers

This preprint studies the expected coprime density of the least common multiple of independent random integers sampled uniformly from {2,…,M}\{2,…,M\}{2,…,M}. For Lk=lcm⁡(X1,…,Xk)L_k=lcm(X_1,…,X_k)Lk=lcm(X1,…,Xk), the central quantity is ρk(M)=E ⁣[φ(Lk)Lk],ρ_k(M)=E\![φ(L_k)/L_k],ρk(M)=E[Lkφ(Lk)], which represents the expected density of integers coprime to LkL_kLk. The paper derives an exact finite-sum identity for ρk(M)ρ_k(M)ρk(M), proves convergence to the Mertens prime product for fixed MMM as k→∞k→→∞, and establishes a sharp geometric asymptotic for the convergence rate. It also analyzes the joint regime k=⌊cM⌋k= cM=⌊cM⌋, M→∞M→→∞, obtaining an explicit limiting function that describes the first-order deviation from the Mertens product. The proof uses a subset expansion over primes, a decomposition by subset cardinality, estimates for elementary symmetric sums, and classical results on reciprocal prime sums. Numerical computations up to M=107M=10^7M=107 illustrate the finite-MMM behavior and support the asymptotic results. The accompanying source package includes the LaTeX manuscript, figures, reproducibility data, and numerical audit files.