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

We apply the generalized monoid of words and logarithmic-prefix functionals of \cite{C1} to the regularization of the divergent product $$P_{\omega }=\prod _{n=1}^{\mathrm{\infty }}{n}^{(-1{)}^{\omega (n)}},$$ where $\omega (n)$ is the number of distinct prime divisors of $n$. The associated Dirichlet series $D_{\omega }(s)={\sum }_{n\ge 1}(-1{)}^{\omega (n)}{n}^{-s}$ has a natural boundary at $\mathrm{\Re }(s)=0$, 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 $D_{\omega }^{\mathrm{\prime }}(0)$ 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 $\alpha =2$, 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 $\zeta (s)$ (de la Vallée Poussin), which yields the effective bound $T_{\omega }(x)=O(x{e}^{-c\sqrt{\log x}})$ for some $c>0$. Numerical computation up to $N={10}^7$ gives the regularized value $$P_{\omega ,\mathrm{r}\mathrm{e}\mathrm{n}}\approx 0.7731,$$ 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 $s=1$, $$D_{\omega }^{\mathrm{\prime }\mathrm{\prime }}(1)=8C_2\ln 2,$$ where $C_2$ is the Hardy–Littlewood twin prime constant, which provides an unconditional analytical link to the arithmetic structure of the sequence $(-1{)}^{\omega (n)}$. A self-contained proof of the natural boundary, based on the Estermann–Dahlquist theorem, is given in Appendix~A.