AI 中文总结
本文在 Pisot 数字系统中证明了固定终端块词的素数定理，解决了 Zeckendorf 猜想，并证明了 Tribonacci 素数定理，同时通过环面模型获得了多项统计性质。
AI 中文摘要
我们证明了 Pisot 整数数字系统中固定终端块词的素数定理。若主导根 $\varphi$ 是 Pisot 数且特征多项式 $P_h$ 是其最小多项式，则每个总数字长度为 $m$ 的终端块词以渐近频率 $\varphi^{-m}$ 出现在素数中。在 Zeckendorf 情形下，这解决了一个最近的猜想。证明将终端条件转化为 Rauzy 柱面窗口，再转化为紧环面上的线性轨道。对于这些伴随替换，所需的单重性 Rauzy 几何是自动的：在反转替换词后，Barge 的纯离散定理适用。Rauzy 环面还给出了替换固定词的素数定理：每个有限因子以素数起始位置出现，且频率为其普通因子频率。这证明了 Drmota--Müllner--Spiegelhofer 提出的 Tribonacci 素数定理。环面模型进一步产生了多项式采样律、与固定同余类的渐近独立性、固定位移相关公式以及 Möbius 正交性。结合已有的素数定理，它给出了 Chebotarev 的终端细化，并表明每个固定的终端素数类包含任意长的算术级数，其公差具有多对数界。
英文摘要
We prove a prime number theorem for fixed terminal block words in Pisot integer numeration systems. If the dominant root $φ$ is a Pisot number and the characteristic polynomial $P_h$ is its minimal polynomial, every terminal block word of total digit-length $m$ occurs among the primes with asymptotic frequency $φ^{-m}$. In the Zeckendorf case this resolves a recent conjecture. The proof converts terminal conditions into Rauzy cylinder windows and then into a linear orbit on a compact torus. For these companion substitutions, the required multiplicity-one Rauzy geometry is automatic: Barge's pure-discreteness theorem applies after reversal of the substitution words. The Rauzy torus also gives a prime number theorem for the substitution fixed word: every finite factor occurs at prime starting positions with its ordinary factor frequency. This proves the Tribonacci prime-number theorem suggested by Drmota--Müllner--Spiegelhofer. The toral model further yields polynomial sampling laws, asymptotic independence from fixed congruence classes, fixed-shift correlation formulas, and Möbius orthogonality. Combined with established prime theorems, it gives a terminal refinement of Chebotarev and shows that every fixed terminal prime class contains arbitrarily long arithmetic progressions with polylogarithmically bounded common difference.
