发表机构
Colégio Pedro II（佩德罗二世学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了带可控无限长度元素的广义字幺半群，将其应用于发散算术乘积的正则化，获得了整数等交替乘积的规范有限值，还扩展了可处理的乘积类型并提出了 Thue-Morse 乘积的猜想。
AI 中文摘要
本文构造了一种广义字幺半群，它是自由幺半群Σ*-的扩展，包含具有可控无限长度的元素。该构造采用双向前缀-后缀度量以及有限字的适度网（moderate nets）上的渐近等价关系，所得商集是一个幺半群，带有自然偏序、长度同态和定义明确的反转对合。在该幺半群上定义了Ecalle的重正分析（resurgent analysis）意义下的 moulds（ mould 是重正分析中的术语，可保留原名）。由渐近等价提供的对数窗口保证了仅依赖对数前缀的 moulds 可降为商集上定义明确的泛函。该框架应用于发散算术乘积的正则化，这类乘积的振荡遵循规则模式。整数、素数和阶乘的交替乘积获得了与ζ正则化（zeta regularization）一致的规范有限值。对称泛函抵消了主导振荡，对数切萨罗重正化（logarithmic Cesaro renormalization）提取了常数项。该方法随后扩展到经典正则化无法处理的乘积，例如符号序列在二进块（dyadic blocks）上为常数的乘积。针对 Thue-Morse 乘积提出了一个猜想，并根据算术序列的发散类型系统化了评估泛函和重正化方案的选择。
英文摘要
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 $\sqrt{2/π}$, matching zeta regularization, while the alternating product of factorials yields $(2/π)^{1/4}$ and satisfies $\mathfrak{F}_{\mathrm{ren}}=\sqrt{P_{\mathrm{ren}}}$. 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 $\exp(\mathcal F)\approx0.9732$. 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.
