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 Pω=∏n=1∞ n(−1)ω(n), where ω(n) is the number of distinct prime divisors of n. The associated Dirichlet series Dω(s)=∑n≥1(−1)ω(n)n−s has a natural boundary at ℜ(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 Dω'(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 Tω(x)=O(x e−c√(log x)) for some c>0. Numerical computation up to N=107 gives the regularized value Pω,ren≈ 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, Dω''(1)=8 C2ln 2, where C2 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.
Related
- Asymptotic monoids of words and logarithmic-prefix functionals for divergent arithmetic products
- Finite Borel asymptotic dimension of bounded-to-one commutative monoid actions
- Asymptotic dimension of commutative monoid actions
- Alternating decomposition of a product in a spherical type Artin-Tits monoid
- The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence
BibTeX
@misc{cruz2026regularization,
title = {{Regularization of a Divergent Arithmetic Product via the Asymptotic Monoid: The Case of $(-1)^ω(n)$}},
author = {A. Alvarez Cruz and E. A. Alvarez Gutierrez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13658},
eprint = {2610.13658},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13658},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Regularization of a Divergent Arithmetic Product via the Asymptotic Monoid: The Case of (−1)^ω(n)
AU - A. Alvarez Cruz
AU - E. A. Alvarez Gutierrez
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13658
UR - https://arxiv.org/abs/2610.13658
AB - We apply the generalized monoid of words and logarithmic-prefix functionals of \cite{C1} to the regularization of the divergent product P_ω=∏_n=1^∞ n^(−1)^ω(n), where ω(n) is the number of distinct prime divisors of n. The associated Dirichlet series D_ω(s)=∑_n≥1(−1)^ω(n)n^−s has a natural boundary at ℜ(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√(log x)) for some c>0. Numerical computation up to N=10^7 gives the regularized value P_ω,ren≈ 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_2ln 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.
ER -