Regularization of a Divergent Arithmetic Product via the Asymptotic Monoid: The Case of $(-1)^{ω(n)}$

Abstract: We apply the generalized monoid of words and logarithmic-prefix functionals of \cite{C1} to the regularization of the divergent product [ P_ω=\prod_{n=1}^{\infty} n^{(-1)^{ω(n)}}, ] where (ω(n)) is the number of distinct prime divisors of (n). The associated Dirichlet series (D_ω(s)=\sum_{n\ge1}(-1)^{ω(n)}n^{-s}) has a natural boundary at (\Re(s)=0), as established by Meretov (1987) and recovered here as a special case of the Estermann--Dahlquist theory of Euler products, which makes zeta regularization impossible: the derivative (D_ω'(0)) does not exist. This product therefore lies completely beyond the reach of classical regularization methods. We construct the canonical net encoding the sign sequence, define the logarithmic-prefix functional with exponent (α=2), and extract a finite value through the weighted evaluation functional of the monoid framework. The evaluation functional is shown to converge absolutely, and this convergence is unconditional: it follows from the classical zero-free region of (ζ(s)) (de la Vallée Poussin), which yields the effective bound (T_ω(x)=O(x\,e^{-c\sqrt{\log x}})) for some (c>0). Numerical computation up to (N=10^7) gives the regularized value [ P_{ω,\mathrm{ren}}\approx 0.7731, ] a new constant of arithmetic origin that cannot be obtained by any classical summation method. We also compute the second derivative of the Dirichlet series at (s=1), [ D_ω''(1)=8\,C_2\ln 2, ] where (C_2) is the Hardy--Littlewood twin prime constant, which provides an unconditional analytical link to the arithmetic structure of the sequence ((-1)^{ω(n)}). A self-contained proof of the natural boundary, based on the Estermann--Dahlquist theorem, is given in Appendix~A.