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

Source 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.