多維度有限型移位的zeta-函數

Abstract

本論文主要研究二維以上有限型移位的 -函數。關於 作用 的 -函數 是由林德推廣阿廷-馬蘇爾 -函數所得到。首先，研究二維的情況。定義跡算子 為在x方向n週期且高度2之花樣的轉移矩陣，此 具有旋轉對稱性。根據 的旋轉對稱性，引進約化跡算子 ，進一步推得 -函數 是一個多項式的無窮乘積的倒數。此外，對於任何由 中的單位模變換決定的傾斜坐標皆可得到相同結果。所以有一族 -函數都是解析函數 的半純擴張，在此我們也研究自然邊界問題。這些 -函數在原點的泰勒級數展開式皆相同，並且其係數皆為整數。因此，可以得到一族在數論上有趣的恆等式。此方法在三維以上的情況也適用，而且可應用到有限範圍交互作用之伊辛模型的熱力學 -函數。

This dissertation investigates zeta functions for d-dimensional shifts of finite type, . A d-dimensional zeta function which generalizes the Artin-Mazur zeta function was given by Lind for action . First, the two-dimensional case is studied. The trace operator which is the transition matrix for x-periodic patterns of period n with height 2 is rotationally symmetric. The rotational symmetry of induces the reduced trace operator . The zeta function is now a reciprocal of an infinite product of polynomials. The results hold for any inclined coordinates, determined by unimodular transformation in . Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function . The natural boundary of zeta function is studied. The Taylor series expansions at the origin for these zeta functions are equal with integer coefficients, yielding a family of identities which are of interest in number theory. The methods used herein are also valid for d-dimensional cases, , and can be applied to thermodynamic zeta functions for the Ising model with finite range interactions.