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.