Asymptotic monoids of words and logarithmic-prefix functionals for divergent arithmetic products
Abstract
We define a generalized monoid of words, an asymptotic quotient of moderate nets of finite words, to regularize divergent products of arithmetic origin. Divergent sequences are encoded as moderate nets over a finite alphabet, while prefix-dependent observables are evaluated along admissible asymptotic scales. We characterize the evaluation and renormalization schemes axiomatically. Once the asymptotic scale and class of admissible regular expansions are fixed, the constant-term prescription is uniquely determined. Thus the regularized value is canonical relative to the chosen scale and evaluation functional, without absolute uniqueness. The regularizing functionals arise as expectations of logarithmic-prefix functionals under empirical distributions of sign oscillations. The alternating product of integers is regularized to √(2/π), matching zeta regularization, while the alternating product of factorials yields (2/π)1/4 and satisfies Fren=√(Pren). Products governed by dyadic block signs acquire finite values through logarithmic-scale averaging in the corresponding dyadic quotient, invariant under dyadic equivalence. For the Thue--Morse sequence, zeta regularization is well defined via meromorphic continuation of its Dirichlet series. A separate convergent functional associated with its logarithmic partial sums yields exp(ℱ)≈0.9732. We also study numerically a substitution-generated sign sequence with exact square-root growth on dyadic blocks and fractal log-periodic structure. Its Dirichlet series shows a candidate log-periodic pole structure, separating local scale constants from a candidate global regularized value. The choice of asymptotic scale is determined by the arithmetic structure of the sequence: logarithmic scales for ordinary alternating products and dyadic scales for block-automatic sequences.
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BibTeX
@misc{cruz2026asymptotic,
title = {{Asymptotic monoids of words and logarithmic-prefix functionals for divergent arithmetic products}},
author = {A. Alvarez Cruz and E. A. Alvarez Gutierrez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.06297},
eprint = {2608.06297},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.06297},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Asymptotic monoids of words and logarithmic-prefix functionals for divergent arithmetic products
AU - A. Alvarez Cruz
AU - E. A. Alvarez Gutierrez
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2608.06297
UR - https://arxiv.org/abs/2608.06297
AB - We define a generalized monoid of words, an asymptotic quotient of moderate nets of finite words, to regularize divergent products of arithmetic origin. Divergent sequences are encoded as moderate nets over a finite alphabet, while prefix-dependent observables are evaluated along admissible asymptotic scales. We characterize the evaluation and renormalization schemes axiomatically. Once the asymptotic scale and class of admissible regular expansions are fixed, the constant-term prescription is uniquely determined. Thus the regularized value is canonical relative to the chosen scale and evaluation functional, without absolute uniqueness. The regularizing functionals arise as expectations of logarithmic-prefix functionals under empirical distributions of sign oscillations. The alternating product of integers is regularized to √(2/π), matching zeta regularization, while the alternating product of factorials yields (2/π)^1/4 and satisfies F_ren=√(P_ren). Products governed by dyadic block signs acquire finite values through logarithmic-scale averaging in the corresponding dyadic quotient, invariant under dyadic equivalence. For the Thue--Morse sequence, zeta regularization is well defined via meromorphic continuation of its Dirichlet series. A separate convergent functional associated with its logarithmic partial sums yields exp(ℱ)≈0.9732. We also study numerically a substitution-generated sign sequence with exact square-root growth on dyadic blocks and fractal log-periodic structure. Its Dirichlet series shows a candidate log-periodic pole structure, separating local scale constants from a candidate global regularized value. The choice of asymptotic scale is determined by the arithmetic structure of the sequence: logarithmic scales for ordinary alternating products and dyadic scales for block-automatic sequences.
ER -