多維度有限型移位的zeta-函數

國立交通大學應用數學系

博 士 論 文

多維度有限型移位的

ζ

-函數

Zeta Functions for Multi-dimensional

Shifts of Finite Type

研究生：胡文貴

指導教授：林松山教授

多維度有限型移位的

ζ

-函數

Zeta Functions for Multi-dimensional

Shifts of Finite Type

研究生：胡文貴 Student：Wen-Guei Hu

指導教授：林松山 Advisor：Song-Sun Lin

國 立 交 通 大 學

應 用 數 學 系

博 士 論 文

A Thesis

Submitted to Department of Applied Mathematics National Chiao Tung University

in partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy in

Applied Mathematics

May 2009

Hsinchu, Taiwan, Republic of China

多維度有限型移位的

ζ

-函數

研 究 生：胡文貴 指導教授：林松山 教授

國 立 交 通 大 學

應 用 數 學 系

摘要

本論文主要研究二維以上有限型移位的

ζ

-函數。關於

]

d

作用

φ

的

ζ

-函數

ζ

0

( )

s

是由林德推廣阿廷-馬蘇爾

ζ

-函數所得到。首先，

研究二維的情況。定義跡算子

T

_n

為在 x 方向 n 週期且高度 2 之花樣的

轉移矩陣，此

T

_n

具有旋轉對稱性。根據

T

_n

的旋轉對稱性，引進約化

跡算子

τ

_n

，進一步推得

ζ

-函數

(

(

)

)

1 1

det

n _n n

I

s

ζ

∞

τ

− =

=

∏

−

是一個

多項式的無窮乘積的倒數。此外，對於任何由

GL ]

₂

( )

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

ζ

-函數都是解

析函數

ζ

0

( )

s

的半純擴張，在此我們也研究自然邊界問題。這些

ζ

-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

Zeta Functions for Multi-dimensional

Shifts of Finite Type

Student：Wen-Guei Hu Advisor：Prof. Song-Sun Lin

Department of Applied Mathematics

National Chiao Tung University

Abstract

This dissertation investigates zeta functions for d-dimensional

shifts of finite type,

d

≥

2

. A d-dimensional zeta function

ζ

0

( )

s

which generalizes the Artin-Mazur zeta function was given by Lind for

d

]

action

φ

. First, the two-dimensional case is studied. The trace

operator

T

_n

which is the transition matrix for x-periodic patterns of

period n with height 2 is rotationally symmetric. The rotational symmetry

of

T

_n

induces the reduced trace operator

τ

_n

. The zeta function

(

)

(

)

1 1

det

n _n n

I

s

ζ

∞

τ

− =

=

∏

−

is now a reciprocal of an infinite product

of polynomials. The results hold for any inclined coordinates, determined

by unimodular transformation in

GL ]

₂

( )

. Therefore, there exists a

analytic function

ζ

0

( )

s

. 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,

d

≥

3

, and can be applied

to thermodynamic zeta functions for the Ising model with finite range

interactions.

誌 謝

在交大總共有六年的時間，碩士班兩年，博士班四年，當初剛來到交大 的時候，沒想到自己會再繼續攻讀博士班，如今能順利拿到了博士學位，內心除 了高興之外，同時也非常感謝許多這一路來幫助我、陪伴我的人。 首先最感謝的就是我的指導教授林松山老師，在研究方面，老師從如何讀書 和讀論文慢慢一步步地教導我，接著老師帶著我做研究，讓我明白做研究的態度 和方法，當我迷網不知所措的時候，總是會適時地指引我方向，鼓勵我繼續往前 邁進，在老師身上，我看到一個學者在研究上的專業、熱情與堅持，是我將來學 習的一個好榜樣；除了研究之外，在做人處世方面，老師也教導我如何應對進退。 老師給我的幫助不是簡短的三言兩語所能言盡的，真心感謝老師這六年來的教 導。 在交大應用數學系這一段時間中，在學業的學習上，系上的老師給了我很多 的幫助，在動態系統與微分方程的學習上，林松山、莊重、石志文、李明佳、林 琦焜、李榮耀老師們教導我很多專業學科上的知識，他們的認真熱心教導，讓我 能夠正確有效地學習，以及王夏聲、許元春老師的實變課程，楊一帆老師的數論 相關課程也讓我受益良多。除了課業之外，系上所有的老師，也曾給我很多協助， 我才能順利地完成學業，謝謝您們。 亦師亦友般的班榮超學長和林吟衡學姐在我研究所這段期間，在研究上給我 非常多的幫助，教導我許多專業知識，樂意與我討論，讓我從中得到很多寶貴的 經驗，在平時，對於學弟妹們都非常照顧，那些情景如今依然歷歷在目；已經畢 業的許正雄、楊智烜、楊定揮學長，雖然相處的時間不多，你們的關懷與照顧， 讓我感覺到溫暖；此外，張志鴻、胡忠澤、楊其儒、陳耀漢學長在學業與生活上 也給我很多的支持與協助；同研究室的昱豪學長、恭儉與明耀和隔壁研究室的明 杰，與你們一起相處的時間非常輕鬆愉快，讓我度過愉快的四年；碩士班的學弟 妹，慧萍、園芳、怡菁、倖綺、玉雯、佳玲、鴻勳、德展、瑛峰和晉育，有了你 們我的生活更加多采多姿。 從小到大，在各方面我的父母總是盡全力支持我，家是我最好的避風港，不 管發生什麼事，都會永遠在那邊守護我，讓我能夠順利完成博士學位；還有我的 哥哥，在我每次回家的時候，都會和我聊聊生活以及學業上的事情，也會鼓勵我 繼續努力向前；再來要感謝我的女朋友園芳(小小)，不管是在我壓力很大或者是 快樂的時候，都會在我身邊陪伴我，妳的陪伴就是我最好的動力來源；胡啵啵是 我家一隻很可愛的巴戈狗，牠總是看起來呆呆可愛的樣子，每次看到牠都會暫時

忘卻煩惱。 碩士班的同學和以前的朋友勝凱、維哲、雅靜、莉君、伯群、佳慧、琬琪， 雖然只能偶而一起聚個餐，不過和你們相處的時光很快樂。其他我沒能一一列舉 到的學長姐、同學、朋友、學弟妹，能與你們認識真的是太好了。 最後，有大家一路上的陪伴幫助，我才能順利地博士班畢業，今後我會繼續 努力勇敢向前邁進。由衷地感謝大家，謝謝你們。 胡文貴 2009\6\30 於交大

Contents

1 Introduction 1

2 Zeta Functions for Two-dimensional Shifts of

Finite Type 10

2.1 Periodic Patterns ……… 10

2.1.1 Ordering matrices and trace operators ………… 11

2.1.2 Rotational matrices……… 18

2.1.3 Periodic patterns ……… 24

2.2 Rationality of

ζ

_n

……… 27

2.3 More Symbols on Larger Lattice ……… 37

2.4 Zeta Functions Presented in Inclined

Coordinates ……… 41

2.5 Analyticity and Meromorphic Extension of

Zeta functions ……… 50

2.5.1 Analyticity of zeta functions ……… 50

2.5.2 Examples ……… 53

2.6 Equations on

]

2

with Numbers in a Finite

Field ……… 59

2.7 Square Lattice Ising Model with Finite Range

Interactions……… 67

3 Zeta Functions for Higher-dimensional Shifts of

Finite Type 73

3.1 Three-dimensional Shifts of Finite Type ……… 73

3.1.1 Periodic patterns, trace operator and

rotational matrices ……… 73

3.1.2 Rationality of

_{, ;}

1 2 12

a a b

ζ

……… 83

3.1.3 Zeta functions in inclined coordinates ………… 91

3.2 Further Results ……… 99

3.2.1 Higher-dimensional shifts of finite type ……… 99

3.2.2 More symbols on larger lattice ……… 100

3.2.3 Three-dimensional Ising Model with

Finite Range Interactions……… 101

1

Introduction

Various zeta functions have been investigated in the fields of number theory, geom-etry, dynamical systems and statistical physics. This work studies the zeta functions in a manner that follows the work of Artin and Mazur [1], Bowen and Lanford [11], Ruelle [45] and Lind [36]. First, recall the zeta function that was defined by Artin and Mazur.

Let φ : X −→ X be a homeomorphism of a compact space and Γn(φ) denote the

number of fixed points of φn_{. The zeta function ζ}

φ(s) for φ defined in [1] is ζφ(s) = exp ∞ X n=1 Γn(φ) n s n ! . (1.1)

Later, Bowen and Lanford [11] demonstrated that if φ is a shift of finite type, then ζφ(s) is a rational function. In the simplest case, when a shift is generated by a

transition matrix A in Z, (1.1) is computed explicitly as

ζA(s) = exp  _∞ P n=1 trAn n s n  (1.2) = (det(I − sA))−1_{, (1.3)} and then ζA(s) = Q λ∈Σ(A) (1− λs)−χ(λ)_, (1.4)

where χ(λ) is a non-negative integer that is the algebraic multiplicity of eigenvalue λ and Σ(A) is the spectrum of A. ζA(s) is a rational function which involves only

eigenvalues of A.

Lind [36] extended (1.1) to Zd_{-action as follows. For Z}d_{-action, d ≥ 1, let φ be an}

action of Zd on X. Denote the set of finite-index subgroups of Zd by Ld. The zeta

function ζφ defined by Lind is

ζφ(s) = exp X L∈Ld ΓL(φ) [L] s [L] ! , (1.5)

where [L] = index[Zd_{/L] and Γ}

L(φ) is the number of fixed points by φn for all n∈ L.

prod-uct formulae, and computed ζφ explicitly for some interesting examples. Furthermore,

he raised some fundamental problems for zeta functions, including the following two. Problem 7.2. [36] For ”finitely determined” Zd_{-actions φ such as shifts of finite}

type, is there a reasonable finite description of ζφ(s)?

Problem 7.5. [36] Compute explicitly the thermodynamic zeta function for the 2-dimensional Ising model, where α is the Z2 _{shift action on the space of configurations.}

The present authors previously studied pattern generation problems in Zd_{, d ≥ 2,}

and developed several approaches such as the use of higher order transition matrices and trace operators to compute spatial entropy [4; 6]. The work of Ruelle [45] and Lind [36] indicated that our methods could also be adopted to study zeta functions.

In this investigation, Problems 7.2 and 7.5 are answered when φ is a shift of finite type. More related results and questions are also addressed. The following paragraphs briefly introduce relevant results.

First, the two dimensional case is studied. Let Z_m×m be the m× m square lattice in Z2 _{and S be the finite set of symbols (alphabets or colors). S}Zm×m _{is the set of}

all local patterns (or configurations) on Z_m×m. A given subset _{B ⊂ S}Zm×m is called

a basic set of admissible local patterns. Σ(B) is the set of all global patterns defined on Z2 _{which can be generated by B. For simplicity, only the results of Z}

2×2 with two

symbols S = {0, 1} are presented here. Subsection 2.3 considers the general case. As presented elsewhere [36],_L2 can be parameterized in Hermite normal form [39]:

L2 =      n l 0 k   Z2 _{: n ≥ 1, k ≥ 1 and 0 ≤ l ≤ n − 1}   .

Given a basic set B, denote by PB

    n l 0 k   

 the set of all   n l

0 k 

-periodic and

B-admissible patterns and ΓB

    n l 0 k     is the number of P_B     n l 0 k    . The zeta function, defined by (1.5), is denoted by

ζ_B0 = exp   ∞ X n=1 ∞ X k=1 n−1 X l=0 1 nkΓB     n l 0 k     snk   . (1.6)

In [36], ζ0

B is shown analytically in |s| < exp(−g(B)), where

g(_{B) ≡ lim sup}

[L]→∞

1

[L]log ΓB(L). (1.7)

In this work, the sum of n and k in (1.6) is treated separately as an iterated sum. Indeed, for any n _{≥ 1, define the n-th order zeta function ζ}n(s) ≡ ζB,n(s) (in

x-direction) as ζn(s) = exp  1 n ∞ X k=1 n−1 X l=0 1 kΓB     n l 0 k     snk   ; (1.8) the zeta function ζ(s)_{≡ ζB}(s) is given by

ζ(s) =

∞

Y

n=1

ζn(s). (1.9)

The first observation of (1.8) is that, for n ≥ 1 and l ≥ 1, any   n l 0 k  -periodic pattern is   n 0 0 nk (n,l) 

-periodic, where (n, l) is the greatest common divisor (GCD) of

n and l. Therefore,  n 0

0 k 

-periodicity of patterns must be investigated in details. The trace operators Tn≡ Tn(B) that were introduced in [6] are useful in studying

  n l

0 k 

-periodic and the B-admissible pattern, where Tn= [tn;i,j] is a 2n×2nmatrix

with tn;i,j ∈ {0, 1}. Tn(B) represents the set of patterns that are B-admissible and

x-periodic of period n with height 2. The trace operator Tn can be used to construct

(doubly) periodic B-admissible patterns. Indeed, for k ≥ 1 and 0 ≤ l ≤ n − 1,

Γ_B     n l 0 k     = tr(Tk nRnl), (1.10)

where Rn is a 2n× 2n rotational matrix defined by

  

R_n;i,2i−1 = 1 and Rn;2n−1_+i,2i = 1 for 1≤ i ≤ 2n−1,

Denote by Rn= n−1_P

l=0

Rl

n; now based on (1.10), ζn(s) becomes

ζn(s) = exp 1 n ∞ X k=1 1 ktr(T k nRn)snk ! , (1.11) which is a generalization of (1.2).

To elucidate the method used to study (1.11), Tnis firstly assumed to be symmetric.

Then Tn can be expressed in Jordan canonical form as

T_n = UJUt (1.12)

where the eigen-matrix U = (U1, ..., UN) is an N× N matrix which consists of linearly

independent (column) eigenvectors Uj, 1 ≤ j ≤ N and N ≡ 2n. Jordan matrix

J = diag(λj) is a diagonal N × N matrix, which comprises eigenvalues λj, 1≤ j ≤ N.

Now, 1 n ∞ P k=1 1 ktr(T k nRn)snk = 1_ntr(U(P∞ k=1 1 kJ k_snk_)Ut_R n) = N P j=1 1 n  Rn◦ UjUjt   log(1 − λjsn)−1 (1.13)

can be proven, where _{◦ is a Hadamard product: if A = [a}i,j]M ×M and B = [bi,j]M ×M,

then A◦ B = [ai,jbi,j]M ×M.

Evaluating the coefficients _|Rn◦ UjUjt| of log(1 − λjsn)−1 is important. Now, the

Rn-symmetry of Tn is crucial. Indeed, let U be an eigenvector of Tn with eigenvalue

λ, then Rl

nU is also eigenvector of Tn for all 0≤ l ≤ n − 1. Notably, Rnn= I2n, where

Im is the m× m identity matrix.

U is called Rn-symmetric, if RlnU = U for all 0 ≤ l ≤ n − 1. And U is called

anti-symmetric ifn−1P

l=0

Rl

nU = 0. Additionally, for any given eigenvalue λ, the associated

eigenspace Eλ can be proven to be spanned by symmetric eigenvectors Uj, 1≤ j ≤ pλ,

and anti-symmetric eigenvectors U′

j, 1 ≤ j ≤ qλ: Eλ = {U1,· · · , Upλ, U1′,· · · , Uq′λ},

where pλ+ qλ = dim(Eλ) and pλ or qλ can be zero.

χ(λ)_≡ 1 n

X

λj=λ

|Rn◦ UjUjt| = pλ (1.14)

is the number of linearly independent symmetric eigenvectors of Tn with respect to λ,

a non-negative integer. Hence, choosing eigen-matrix U in (1.12), which consists of symmetric and anti-symmetric eigenvectors, yields

ζn(s) =

Y

λ∈Σ(Tn)

(1_{− λs}n)−χ(λ) (1.15) as a rational function, as in (1.4).

To further study χ(λ) in (1.14), the reduced trace operator τn is introduced as

follows. From the rotational matrix Rn, for 1 ≤ i ≤ 2n, the equivalent class Cn(i) of

i is defined as Cn(i) =

n j Rl

n

i,j = 1 for some 1≤ l ≤ n

o

. The index set In of n is

defined by _In =



i1 ≤ i ≤ 2n_{, i≤ j for all j ∈ C} n(i)

and χn is the cardinal number

of _In. Indeed, χn is the number of necklaces that can be made from n beads of two

colors when the necklaces can be rotated but not turned over. Furthermore,

χn = 1 n X d|n φ(d)2n/d, (1.16)

where φ(d) is the Euler totient function.

Then, the reduced trace operator τn = [τn;i,j] of Tn is a χn× χn matrix that is

defined by

τn;i,j =

X

k∈Cn(j)

tn;i,k (1.17)

for each i, j ∈ In. λ∈ Σ(Tn) with χ(λ) ≥ 1 can be verified if and only if λ ∈ Σ(τn).

Moreover, χ(λ) is the algebraic multiplicity of τn with eigenvalue λ. Therefore,

ζn(s) = (det (I − snτn))−1, (1.18)

a similar formula as in (1.3). Hence, the zeta function ζ(s) is obtained as

ζ(s) =

∞

Y

n=1

which is an infinite product of rational functions. Equation (1.19) generalizes (1.3) and is a solution to Lind’s Problem 7.2. Furthermore, according to (1.19), the coefficients of Taylor series expansion for ζ(s) at s = 0 are integers, as obtained by Lind [36].

As presented elsewhere [6], an another trace operator bTn is B-admissible and

y-periodic of period n with width 2 along the x-axis. Indeed, _L2 can be parameterized

as another Hermite normal form, and n-th order zeta function bζn(s) is defined by

b ζn(s) = exp  1 n ∞ X k=1 n−1 X l=0 1 kΓB     k 0 l n     snk   , (1.20) and the zeta function bζ(s) is defined by

b ζ(s) = ∞ Y n=1 b ζn(s). (1.21) Therefore, b ζ(s) = ∞ Y n=1 Y λ∈Σ( bTn) (1− λsn₎−bχ(λ) _(1.22) = ∞ Y n=1 (det (I _{− s}nτ_bn))−1. (1.23)

The construction of the zeta functions ζ and bζ in rectangular coordinates can be extended to an inclined coordinates system. Indeed, let the unimodular transformation

γ be an element of the unimodular group GL2(Z): γ =

  a b

c d 

, a, b, c and d are integers and ad− bc = ±1. The lattice Lγ is defined by

Lγ ≡   n l 0 k   γ Z2 ₌   na la + kc nb lb + kd   Z2_{. (1.24)}

The n-th order zeta function of ζ_B0(s) with respect to γ is defined by

ζ_B;γ;n(s) = exp  1 n ∞ X k=1 n−1 X l=0 1 kΓB     n l 0 k   γ   snk   , (1.25) and the zeta function ζ_B;γ with respect to γ is given by

ζ_B;γ(s)≡

∞

Y

n=1

The n-th order rotational matrix Rγ;n, trace operator Tγ;n(B) and reduced trace

oper-ator τγ;n(B) can also be introduced and

ζ_B;γ;n(s) = (det (I _{− s}n_τ

γ;n))−1. (1.27)

Therefore, the zeta function ζ_B;γ is given by

ζ_B;γ(s) = ∞ Y n=1 (det(I _{− s}n_τ γ;n))−1. (1.28)

Since the iterated sum in (1.25) and (1.26) is a rearrangement of ζ0 B(s),

ζ_B;γ(s) = ζ_B0(s) (1.29)

for |s| < exp(−g(B)). The identity (1.29) yields a family of identities when ζB;γ is

expressed as a Taylor series expansion at the origin s = 0. The further applications of these identities in number theory will appear elsewhere.

Note that, one may consider the zeta functions ζr

B, which only involves

  n 0

0 k  -periodic patterns, defined by

ζ_Br = exp  X∞ n=1 ∞ X k=1 1 nkΓB     n 0 0 k     snk   . (1.30) However, in general, for n_{≥ 1, χ(λ) is not an integer in (1.15) for ζ}r

B;n and ζB;nr is not

a rational function. Therefore, ζr

B is not an infinite product of rational functions and

may lose some important properties such as GL2(Z) invariant.

The thermodynamic zeta function raised by Ruelle [45] with weight function θ : X_{→ (0, ∞) was defined by Lind [36] as}

ζ0 α,θ(s) = exp  X L∈Ld    X x∈fixL(α) Y k∈Zd_/L θ αk_x
   s[L] [L]   , (1.31) where f ixL(α) is the set of points fixed by αn for all n∈ L.

For the Ising model, where α is a shift of finite type given by B and the weight function θ is a potential with finite range, the previous arguments apply. Indeed, the trace operator TIsing;n(B) and reduced trace operator τIsing;n(B) can be defined, and

ζ_Ising;B(s) =

∞

Y

n=1

(det (I_{− s}nτIsing;n))−1. (1.32)

Equation (1.32) is a solution of Lind’s Problem 7.5. Furthermore, the relations of critical phenomenon in phase transition with the zeta functions will be investigated later.

Notably, the methods herein also apply to sofic shifts. The results will appear elsewhere.

It is clear that in many situations the three-dimensional problems are more related to our real

world phenomena. Now, the zeta functions of d-dimensional shifts of finite type are studied for

d ≥ 3, and the previous results of Z2 _{are extended. For simplicity, only the zeta}

functions for three-dimensional shifts of finite type are introduced and the general case is studied in Subsection 3.2.

Let Z_m×m×m be the m_{×m ×m cubic lattice in Z}3 _{andS be the finite set of symbols}

(alphabets or colors). _SZm×m×m is the set of all local patterns on Z

m×m×m. Denote

B ⊂ SZm×m×m _{a basic set of admissible local patterns and} P(B) the set of all periodic

patterns that are generated by _{B on Z}3_.

The Hermite normal form [39] can be used to parameterize _L3 as

L3 =               a1 b12 b13 0 a2 b23 0 0 a3     Z 3 _{: a} i ≥ 1, 1 ≤ i ≤ 3, 0 ≤ bij ≤ ai− 1, i + 1 ≤ j ≤ 3          .

Given a basic set_{B. Let L =}      a1 b12 b13 0 a2 b23 0 0 a3     Z 3 _{∈ L} 3, denotePB           a1 b12 b13 0 a2 b23 0 0 a3           the set of all L-periodic patterns that are generated by _{B on Z}3 _and

Γ_B           a1 b12 b13 0 a2 b23 0 0 a3           the number of PB           a1 b12 b13 0 a2 b23 0 0 a3         

. Then, the zeta func-tion in (1.1) is

ζ_B0 = exp      3 X i=1 ∞ X ai=1 3 X j=i+1 a_i−1 X bij=0 1 a1a2a3 Γ_B           a1 b12 b13 0 a2 b23 0 0 a3          s a1a2a3     . (1.33) Similar to (1.8) and (1.9), the (a1, a2; b12)-th zeta function is defined by

ζ_B;a1,a2;b12(s) = exp      1 a1a2 ∞ X a3=1 a_X1−1 b13=0 a_X2−1 b23=0 1 a3 Γ_B           a1 b12 b13 0 a2 b23 0 0 a3          s a1a2a3      (1.34) and the zeta function ζ_B(s) is given by

ζ_B(s) = ∞ Y a1=1 ∞ Y a2=1 a_Y1−1 b12=0 ζ_B;a1,a2;b12(s). (1.35)

The trace operator Ta1,a2;b12(B) and rotational matrices Rx;a1,a2;b12 and

Ry;a1,a2;b12 are introduced. After the rotational symmetry of Ta1,a2;b12 is demonstrated

the reduced trace operator τa1,a2;b12(B) can be defined. Finally, as in (1.18), ζB;a1,a2;b12(s)

can be represented as a rational function:

ζ_B;a1,a2;b12(s) = (det (I − s a1a2_τ a1,a2;b12)) −1_{. (1.36)} Hence, ζ_B(s) = ∞ Y a1=1 ∞ Y a2=1 aY1−1 b12=0 (det (I − sa1a2_τ a1,a2;b12)) −1 _(1.37)

is a reciprocal of an infinite product of polynomials. Here, we show (1.36) by using a simpler and more straightforward method than that for (1.18). However, the proof of (1.18) is also valid for d_{≥ 3.}

Additionally, for any γ ∈ GL3(Z), the zeta function can also be represented in

γ-coordinates. Therefore, a family of zeta functions exists that have the same integer coefficients in their Taylor series expansions at s = 0.

As in (1.32), the thermodynamic zeta function for the three-dimensional Ising model with finite range interactions can also be represented as a reciprocal of an infinite

product of polynomials. The dimensional model can be applied to study three-dimensional phase-transitions problems. The further results need to be investigated.

Some references that are related to our work are listed here. Zeta functions and related topics [1; 5; 11; 20; 22; 23; 24; 30; 31; 36; 37; 38; 40; 41; 42; 44; 45; 47]; patterns generation problems and lattice dynamical systems [2; 3; 4; 6; 7; 8; 12; 13; 14; 15; 16; 17; 18; 19; 25; 26; 28; 29; 34; 35], and phase-transitions in statistical physics [9; 10; 32; 33; 43] have all been covered elsewhere.

The rest of this dissertation is organized as follows. In Section 2, the trace operator Tn(B) and rotational matrix Rn are introduced to accommodate the periodic patterns.

Based on the rotational symmetry of the trace operator, the reduced trace operator τn(B) is defined. Therefore, the rationality of ζB;n is obtained. The results also hold

when inclined coordinates are used for any unimodular transformation γ _{∈ GL}2(Z).

The meromorphic extension of zeta function is studied. The zeta function of the solution set of equations on Z2 _{with numbers from a finite field is also investigated.}

Finally, the method is applied to thermodynamic zeta function for the square Ising model with a finite range interactions.

In Section 3, the three-dimensional case is studied first. The trace operator Ta1,a2;b12

and rotational matrices Rx;a1,a2;b12 and Ry;a1,a2;b12 are introduced to study periodic

pat-terns. The rotational symmetry of Ta1,a2;b12 induces the reduced trace operator τa1,a2;b12

and then the rationality of

ζ_B;a1,a2;b12 is obtained. The results hold for any inclined coordinates, determined by

unimodular transformation in GL3(Z). Finally, the d-dimensional cases, d ≥ 4, and

thermodynamic zeta functions for the three-dimensional Ising model with finite range interactions are studied.

2

Zeta functions for two-dimensional shifts of finite type

In this section, zeta functions for two-dimensional shifts of finite type are studied.

2.1 Periodic patterns

This subsection first reviews the ordering matrices of local patterns and trace operators [4; 6]. It then derives rotational matrices Rn and Rn, and studies their properties.

The Rn-symmetry of the trace operator is also discussed. Finally, some properties of

periodic patterns in Z2 _{are investigated. In particular, the}

  n l 0 k  -periodic pattern is proven to be   n 0 0 _(n,l)nk  -periodic.

For clarity, two symbols on the 2_{× 2 lattice Z2×2} are initially examined. Subsection 2.3 addresses more general situations.

2.1.1 Ordering matrices and Trace operators

For given positive integers N1 and N2, the rectangular lattice ZN1×N2 is defined by

Z_N

1×N2 ={(n1, n2)|0 ≤ n1 ≤ N1− 1 and 0 ≤ n2 ≤ N2− 1} .

In particular, Z_2×2 = {(0, 0), (1, 0), (0, 1), (1, 1)}. Define the set of all global patterns on Z2 _{with two symbols {0, 1} by}

Σ2₂ ={0, 1}Z2 ₌_{U|U : Z}2 _{→ {0, 1} .}

Here, Z2 _={(n

1, n2)|n1, n2 ∈ Z}, the set of all planar lattice points (vertices). The set

of all local patterns on ZN1×N2 is defined by

ΣN1×N2 ={U|Z_N1×N2 : U ∈ Σ

2 2}.

Now, for any givenB ⊂ Σ2×2,B is called a basic set of admissible local patterns. In

short, _{B is a basic set. An N}1× N2 pattern U is called B-admissible if for any vertex

(lattice point) (n1, n2) with 0≤ n1 ≤ N1− 2 and 0 ≤ n2 ≤ N2− 2, there exists a 2 × 2

admissible pattern (βk1,k2)0≤k1,k2≤1 ∈ B such that

Un1+k1,n2+k2 = βk1,k2,

for 0_{≤ k}1, k2 ≤ 1. Denote by ΣN1×N2(B) the set of all B-admissible patterns on ZN1×N2.

As presented elsewhere [4], the ordering matrices X_2×2 and Y_2×2 are introduced to arrange systematically all local patterns in Σ_2×2.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . (2.1)

The vertical ordering matrix Y_2×2 = [yi,j]4×4 is defined by

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 . (2.2)

It is clear that the local pattern yi,j in Y2×2 is the reflection π4 of xi,j in X2×2, i.e,

. The reflection can be represented by   0 1 1 0   in GL2(Z) with determinant −1.

In (2.1) and (2.2), the orders of the pattern

β0,1 β1,1

β0,0 β1,0, β_i,j ∈ {0, 1}, are given by

2 4 1 3 and

3 4

1 2 respectively. X_2×2 and Y_2×2 are clearly related as follows.

X_2×2=        y1,1 y1,2 y2,1 y2,2 y1,3 y1,4 y2,3 y2,4 y3,1 y3,2 y4,1 y4,2 y3,3 y3,4 y4,3 y4,4        (2.3)

and Y_2×2 =        x1,1 x1,2 x2,1 x2,2 x1,3 x1,4 x2,3 x2,4 x3,1 x3,2 x4,1 x4,2 x3,3 x3,4 x4,3 x4,4        . (2.4)

The set C_2×2 = [ci,j], which consists of all x-periodic patterns of period 2 with height

2 can be constructed from Y_2×2 as follows.

0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 C_2×2= . (2.5)

The patterns in C_2×2 are expressed as elements in Σ_3×2 and are understood to be extendable periodically in the x-direction to all of Z_∞×2. Notably,

   c1,2 ∼= c1,3, c2,1 ∼= c3,1, c2,2 ∼= c3,3, c2,3 ∼= c3,2, c2,4 ∼= c3,4, c4,2 ∼= c4,3, (2.6)

where ci,j ∼= ci′_,j′ means that c_i′_,j′ is an x-translation by one step from c_i,j. Later, the

translation invariance property (2.6) will be shown to imply R2-symmetry of the trace

operator T2.

Finally, P_2×2 denotes the set of   2 0

0 2 

-periodic patterns, which can be recorded from C_2×2 or Y_2×2 as an element in Σ_3×3 as follows.

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 P_2×2= . _(2.7)

Notably, the upper two rows from the top of each pattern in P_2×2 is Ct

2×2, where

Ct_2×2 is the transpose of C_2×2.

Therefore, P_2×2 can be regarded as a ”Hadamard type product •” of C2×2 with

Ct_2×2, given by the following construction.

P_2×2 = C_{2×2• C}t_2×2; (2.8) the lower two rows of each pattern in P_2×2 come from C_2×2, and the upper two rows come from Ct

2×2; they are glued together by the middle row. Equation (2.8) is the

prototype for constructing doubly periodic patterns of Z2 _{from x-periodic patterns.}

Later, this idea will be generalized to all doubly periodic patterns.

The y-ordering matrices of patterns in Σ_n×2, n≥ 2, can be ordered analogously by

Y_n×2 = [yn;i,j] = β0,1 β1,1 · · · βn−1,1 β0,0 β1,0· · · βn−1,0 ₂n × 2n , _(2.9) where    i = ψ(β0,0β1,0· · · βn−1,0), j = ψ(β0,1β1,1· · · βn−1,1), (2.10)

and the n-th order counting function ψ _{≡ ψ}n:{0, 1}Zn → {j|1 ≤ j ≤ 2n} is defined by

ψ(β0β1· · · βn−1) = 1 + n−1

X

j=0

βj2(n−1−j). (2.11)

The recursive formulas for generating Y_n×2 from Y_2×2, taken from another investi-gation [4], is as follows.

Let Y_n×2=   Yn×2;1 Yn×2;2 Y_n×2;3 Y_n×2;4   , (2.12)

where Y_n×2;i is a 2n−1_{× 2}n−1 _{matrix of patterns. Then,}

Y_(n+1)×2 =        x1,1Yn×2;1 x1,2Yn×2;2 x2,1Yn×2;1 x2,2Yn×2;2 x1,3Yn×2;3 x1,4Yn×2;4 x2,3Yn×2;3 x2,4Yn×2;4 x3,1Yn×2;1 x3,2Yn×2;2 x4,1Yn×2;1 x4,2Yn×2;2 x3,3Yn×2;3 x3,4Yn×2;4 x4,3Yn×2;3 x4,4Yn×2;4        (2.13) is a 2n+1_{× 2}n+1 _matrix.

Hence, x-periodic patterns of period n with height 2 can be expressed in Σ_(n+1)×2, and recorded as an element in C_n×2 by

C_n×2 = β0,1 β0,1 β1,1 · · · βn−1,1 β0,0 β0,0 β1,0 · · · βn−1,0 ₂n_{× 2}n , (2.14) where βi,j ∈ {0, 1}.

Now, given any basic set B, define the associated horizontal and vertical transition matrices H2 = H2(B) = [ap,q] and V2 = V2(B) = [bi,j] by ap,q=    1 if xp,q ∈ B, 0 if xp,q ∈ B,/ and bi,j =    1 if yi,j ∈ B, 0 if yi,j ∈ B,/ (2.15) respectively. Then, H2 =        a1,1 a1,2 a1,3 a1,4 a2,1 a2,2 a2,3 a2,4 a3,1 a3,2 a3,3 a3,4 a4,1 a4,2 a4,3 a4,4        =        b1,1 b1,2 b2,1 b2,2 b1,3 b1,4 b2,3 b2,4 b3,1 b3,2 b4,1 b4,2 b3,3 b3,4 b4,3 b4,4        , (2.16) and V2 =        b1,1 b1,2 b1,3 b1,4 b2,1 b2,2 b2,3 b2,4 b3,1 b3,2 b3,3 b3,4 b4,1 b4,2 b4,3 b4,4        =        a1,1 a1,2 a2,1 a2,2 a1,3 a1,4 a2,3 a2,4 a3,1 a3,2 a4,1 a4,2 a3,3 a3,4 a4,3 a4,4        . (2.17)

The associated column matrices eH₂ of H2 and eV2 of V2 are defined as e H₂ =        a1,1 a2,1 a2,1 a2,2 a3,1 a4,1 a3,2 a4,2 a1,3 a2,3 a1,4 a2,4 a3,3 a4,3 a3,4 a4,4        (2.18) and e V2 =        b1,1 b2,1 b2,1 b2,2 b3,1 b4,1 b3,2 b4,2 b1,3 b2,3 b1,4 b2,4 b3,3 b4,3 b3,4 b4,4        , (2.19) respectively.

The trace operators T2 = T2(B) and bT2 = bT2(B) which were introduced in [6] are

defined as

T₂ = V2◦ eH2 and bT2 = H2◦ eV2, (2.20)

where ◦ is the Hadamard product: if A = [αi,j]p×p and B = [βi,j]p×p, then A◦ B =

[αi,jβi,j]p×p. More precisely,

T2 = [ti,j]22_×22 =        a1,1a1,1 a1,2a2,1 a2,1a1,2 a2,2a2,2 a1,3a3,1 a1,4a4,1 a2,3a3,2 a2,4a4,2 a3,1a1,3 a3,2a2,3 a4,1a1,4 a4,2a2,4 a3,3a3,3 a3,4a4,3 a4,3a3,4 a4,4a4,4        (2.21) and b T2 =  bti,j  22_×22 =        b1,1b1,1 b1,2b2,1 b2,1b1,2 b2,2b2,2 b1,3b3,1 b1,4b4,1 b2,3b3,2 b2,4b4,2 b3,1b1,3 b3,2b2,3 b4,1b1,4 b4,2b,24 b3,3b3,3 b3,4b4,3 b4,3b3,4 b4,4b4,4        . (2.22)

ti,j =

  

1 if ci,j is B-admissible,

0 if ci,j is not B-admissible,

(2.23)

where ci,j ∈ C2×2.

Therefore, T2 is the transition matrix of the B-admissible and x-periodic patterns

of period 2 with height 2. Similarly, bT₂ is the transition matrix of _{B-admissible and} y-periodic patterns of period 2 with width 2.

The translation invariance property (2.6) of C_2×2 implies the following symmetry of T2;    t1,2 = t1,3, t2,1 = t3,1, t2,2 = t3,3, t2,3 = t3,2, t2,4 = t3,4, t4,2 = t4,3. (2.24)

The symmetry of (2.6) or (2.24) can also be identified as the rotational symmetry of a cylinder since elements in C_2×2 can be regarded as cylindrical patterns.

The recursive formulas of Y_n×2 can also be applied to Vn. Indeed, if

Vn=   Vn;1 Vn;2 Vn;3 Vn;4   2n_×2n ,

where Vn;j is a 2n−1× 2n−1 matrix, then

Vn+1 =        a1,1Vn;1 a1,2Vn;2 a2,1Vn;1 a2,2Vn;2 a1,3Vn;3 a1,4Vn;4 a2,3Vn;3 a2,4Vn;4 a3,1Vn;1 a3,2Vn;2 a4,1Vn;1 a4,2Vn;2 a3,3Vn;3 a3,4Vn;4 a4,3Vn;3 a4,4Vn;4        (2.25) and Tn = Vn◦            E2n−2 ⊗   a1,1 a2,1 a3,1 a4,1   E2n−2 ⊗   a1,2 a2,2 a3,2 a4,2   E2n−2 ⊗   a1,3 a2,3 a3,3 a4,3   E2n−2 ⊗   a1,4 a2,4 a3,4 a4,4              , (2.26)

Now, Tn represents the transition matrix of B-admissible x-periodic patterns of

period n with height 2. Similarly, bTn represents the transition matrix of B-admissible

y-periodic patterns of period n with width 2.

2.1.2 Rotational matrices

In this subsection, the rotational matrices Rnand the invariant property of Cn×2under

Rn are investigated and the Rn-symmetry of Tn is then proven.

The shift of any n-sequence β = (β0β1· · · βn−2βn−1), n ≥ 2, βj ∈ {0, 1}, is defined

by

σ((β0β1· · · βn−2βn−1))≡ σn((β0β1· · · βn−2βn−1)) = (β1β2· · · βn−1β0). (2.27)

The subscript of σn is omitted for brevity. Notably, the shift (to the left) of any

one-dimensional periodic sequence (β0β1· · · βn−1β0· · · ) of period n becomes

(β1β2· · · βn−1β0β1· · · ).

The 2n_{× 2}n _{rotational matrix R}

n = [Rn;i,j], Rn;i,j ∈ {0, 1}, is defined by

Rn;i,j = 1 if and only if

i = ψ(β0β1· · · βn−1) and j = ψ(σ(β0β1· · · βn−1)) = ψ(β1β2· · · βn−1β0). (2.28)

From (2.28), for convenience, denote by

j = σ(i). (2.29)

Clearly, Rn is a permutation matrix: each row and column of Rn has one and only

one element with a value of 1. Indeed, Rn can be written explicitly as follows, the

proof is omitted.

Lemma 2.1   

R_n;i,2i−1 = 1 and Rn;2n−1_+i,2i = 1 for 1≤ i ≤ 2n−1,

Rn;i,j = 0 otherwise,

or equivalently, σ(i)_{≡ σ}n(i) =    2i− 1 for 1≤ i ≤ 2n−1_, 2(i− 2n−1_{) for 1 + 2}n−1_{≤ i ≤ 2}n_. (2.31) Furthermore, Rn

n = I2n and for any 1≤ j ≤ n − 1,

(R_nj)i,σj_(i) = 1. (2.32)

The equivalent class Cn(i) of i is defined by

Cn(i) = {σj(i)|0 ≤ j ≤ n − 1}

= nj Rl n

i,j = 1 for some 1≤ l ≤ n

o .

(2.33)

Clearly, either Cn(i) = Cn(j) or Cn(i)∩ Cn(j) =∅. Let i be the smallest element in its

equivalent class, and the index set _In of n is defined by

In= {i|1 ≤ i ≤ 2n, i≤ σq(i), 1≤ q ≤ n − 1} = i1 ≤ i ≤ 2n_{, i≤ j for all j ∈ C} n(i) . (2.34)

Therefore, for each n _{≥ 1, {j|1 ≤ j ≤ 2}n_{} = ∪} i∈In

Cn(i). The cardinal number of In is

denoted by χn. Notably, χn can be identified as the number of necklaces that can be

made from n beads of two colors, when the necklaces can be rotated but not turned over [48]. Moreover, χn is expressed as

χn = 1 n X d|n φ(d)2n/d, (2.35)

where φ(n) is the Euler totient function, which counts the numbers smaller or equal to n and prime relative to n,

φ(n) = nY p|n  1− 1 p  . (2.36)

Example 2.2 Rn, In and Cn(i) for n = 2 and 3, (i) R2=        1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1        , and              C2(1) = {1}, 1→ 1, C2(2) = C2(3) ={2, 3}, 2 → 3 → 2, C2(4) = {4}, 4→ 4, I2 ={1, 2, 4}. (ii) For R3,                    1_{→ 1,} 2_{→ 3 → 5 → 2,} 4→ 7 → 6 → 4, 8_{→ 8,} I3 ={1, 2, 4, 8}.

The following proposition shows the permutation character of Rn and the proof is

omitted.

Proposition 2.3 Let M = [Mi,j]2n_×2n be a matrix where M_i,j denotes a number or a

pattern or a set of patterns. Then,

(RnM)i,j = Mσ(i),j and (MRn)i,j = Mi,σ−1_(j). (2.37)

Furthermore, for any l_{≥ 1,}

(R_nlM)i,j = Mσl_(i),j and (MR_nl)_i,j = M_i,σ−l_(j). (2.38)

In the following, x-periodic patterns of period n with height k_{≥ 1 are studied. More} notation is required.

Definition 2.4

(i) For any n _{≥ 1, let (β}0β1· · · βn−1)∞ be a periodic sequence of period n, denoted by

β = (β0· · · βn−1). σ(β) = σ((β0β1· · · βn−1)) = (β1β2· · · βn−1β0). For any fixed n ≥ 1

(ii) For fixed n≥ 1 and any k ≥ 1, denote by [β0β1· · · βk−1] = (β0,0β1,0· · · βn−1,0)∞⊕ · · · ⊕ (β0,k−1β1,k−1· · · βn−1,k−1)∞ = β0,0 β0,0 β1,0 βn−1,0 β0,1 β0,1 β1,1 βn−1,1 β0,k−1 β0,k−1 β1,k−1 βn−1,k−1 ,

a x-periodic pattern of period n with height k.

(iii) A Hadamard type product _{• of patterns is defined as follows.} [β₀β₁]_{• [β1}β₂] = [β₀β₁β₂]

and

[β₀β_{1· · · βk−1}] = [β₀β₁]_{• [β1}β₂]_{• · · · • [βk−2}β_k−1]. (iv) A 2n_{× 2}n _{ordering matrix C}

n×k = [Cn×k;i,j] of x-periodic patterns of period n with

height k_{≥ 2 is defined by}

C_n×k;i,j ={[β0β1· · · βk−1]|ψ(β0) = i and ψ(βk−1) = j}.

(v) For n _{≥ 1 and k ≥ 2, denote by D}n,k = [Dn,k;i,j] the ordering matrix of patterns,

which consists of a first row β₀ and the k-th row β_k−1 of C_n×k:

Dn,k;i,j ={[β0βk−1]|[β0β1· · · βk−1]∈ Cn×k;i,j}.

Some remarks should be made.

Remark 2.5

(1) For any n≥ 1, the length of β in (i) and βj in (ii) depends on n. For simplicity,

these dependencies are omitted.

(2) The product _{• defined in (iii) applies only when the top row of the first pattern is} identical to the first row of the second pattern.

(3) In (iv), when k = 2, (2.14) applies.

(4) C_n×k;i,j is a set of patterns with the same first and k-th rows. Dn,k is exactly Cn×2,

but, importantly, in C_n×k, all patterns in the entry C_n×k;i,j have the same top and first rows, which can be used to construct y-periodic patterns with a shift in the (k+1)-th row.

In the following lemma, Rn is used to shift the first row in Dtn,k.

Lemma 2.6 Let i = ψ(β₀) and j = ψ(β_k−1). Then (i) (RnDtn,k)i,j = [β_k−1σ(β0)],

(ii) (C_n×k• RnDtn,k)i,j = [β0β1· · · βk−1σ(β0)].

Proof. (i) follows easily from Proposition 2.3 and part (v) of Definition 2.4. From parts (i) and (iii) of Definition 2.4, a product in (ii) is legitimate since the top row of C_n×k and the first row of RnDtn,k are βk−1, and (ii) follows from (i).

Furthermore, the following result shows that the patterns in C_{n×k• R}l

nDtn,k are the

same as the patterns in diag(C_n×(k+1)Rn−l

n ) where diag(M) is the diagonal part of M,

such that diag(M) = I ◦ M. They are important in constructing y-periodic patterns. Proposition 2.7 For any n≥ 2, k ≥ 1 and 0 ≤ l ≤ n,

patterns in C_{n×k• R}l

nDtn,k = patterns in diag(Cn×(k+1)Rn−ln )

= {[β0· · · βk−1σl(β0)]|[β0· · · βk−1]∈ Cn×k}.

Proof. By (2.38), for any 0_{≤ l ≤ n − 1, 1 ≤ i, j ≤ 2}n+1_,

(C_n×(k+1)Rn−ln )i,j ={[β0· · · βk−1σl−n(βk)] : ψ(β0) = i and ψ(βk) = j}.

Since ψ(β_k) = ψ(β₀) = i implies β_k= β₀,

(C_n×(k+1)Rn−l

n )i,i ={[β0· · · βk−1σl−n(β0)] : ψ(β0) = i}.

However, for any 1 _{≤ i, j ≤ 2}n_{, part (ii) of Lemma 2.6 implies}

(C_{n×k• R}l

nDn,kt )i,j = [β0β1· · · βk−1σl(β0)].

Now, for any 0_{≤ l ≤ n − 1 and β = (β}0· · · βn−1),

The proof is complete.

The rotational symmetry of Tn is determined by studying Cn×2 in more detail.

Given a basic admissible set_{B ⊂ Σ2×2}, Tn is defined by (2.26). Let [β0β1]∈ Cn×2, for

0≤ j ≤ n − 1, denote

pj = 2βj,0+ βj,1+ 1,

then the associated entry in Tn is

Tn([β0β1])≡ ap0,p1ap1,p2· · · apn−1,p0. (2.39)

[β₀β₁] is _{B-admissible if and only if a}pj,pj+1 = 1 for all 0≤ j ≤ n − 1, where pn = p0.

Theorem 2.8 For any n _{≥ 2, the trace operator T}n = [tn;i,j]2n_×2n has the following

Rn-symmetry:

tn;σl_(i),σl_(j)= t_n;i,j (2.40)

for all 1_{≤ i, j ≤ 2}n _{and 0≤ l ≤ n − 1.}

Proof. Given [β0β1] ∈ Cn×2, all [σl(β0)σl(β1)], 0 ≤ l ≤ n − 1, represent similar

x-periodic patterns. The entry of [σl_(β

0)σl(β1)] in Tn is

Tn([σl(β0)σl(β1)]) = apl,pl+1apl+1,pl+2· · · apn−1,p0ap0,p1· · · apl−1,pl. (2.41)

Comparing (2.39) with (2.41) clearly reveals that

Tn([β0β1]) = Tn([σl(β0)σl(β1)]) (2.42)

for all 0 ≤ l ≤ n − 1. Additionally, if Tn = [tn;i,j] with i = ψ(β0) and j = ψ(β1), then

(2.42) implies

tn;σl_(i),σl_(j)= t_n;i,j for all 0≤ l ≤ n − 1.

The proof is complete.

Proposition 2.7 and Theorem 2.8 yield the following theorem.

Theorem 2.9 For any n≥ 2 and k ≥ 2, 0 ≤ l ≤ n − 1, |Tk−1

n ◦ RlnTtn| = tr(TknRn−ln ) (2.43)

and

|Tk−1

where Rn= n−1 X l=0 R_nl. (2.45)

Proof. From Proposition 2.7, (2.39) and the properties of Tn, (2.43) follows. Equations

(2.43) and (2.45) yield (2.44). The proof is complete.

2.1.3 Periodic patterns

This subsection studies in detail (double) periodic patterns in Z2_{. Indeed, consider a}

lattice L with Hermite normal form,

L =   n l 0 k   Z2_{, (2.46)}

where n≥ 1, k ≥ 1 and 0 ≤ l ≤ n − 1. A pattern U = (βi,j)i,j∈Z is called L-periodic if

every i, j _{∈ Z} βi+np+lq,j+kq= βi,j (2.47) for all p, q _{∈ Z.} The periodicity of   n l 0 k   and   n 0 0 k′ 

 are closely related as follows.

Proposition 2.10 For any n _{≥ 2, k ≥ 1 and 0 ≤ l ≤ n − 1,}   n l 0 k  -periodic patterns are   n 0 0 _(n,l)nk 

-periodic where (n, l) is the greatest common divisor (GCD) of n and l. Proof. By (2.47), the   n l 0 k 

-periodic pattern is easily identified as 

 n l· m 0 k_{· m}

 -periodic for all m_{∈ N. By taking m =} n

(n,l), the result holds.

Given an admissible set B ⊂ Σ2×2, defined on square lattice Z2×2, the periodic

lattice with the left-bottom vertex (i, j):

Z

2×2((i, j)) ={(i, j), (i + 1, j), (i, j + 1), (i + 1, j + 1)} .

Now, the admissibility is demonstrated to have to be verified on finite square lattices.

Proposition 2.11 An L-periodic pattern U is_{B-admissible if and only if}

U_Z

2×2((i,j)) ∈ B (2.48)

for any 0_{≤ i ≤ n − 1 and 0 ≤ j ≤ k − 1.}

Proof. The proof follows easily from (2.47). The details are left to the reader. According to Proposition 2.11, the admissibility of U is determined by

(βi,j)0≤i≤n,0≤j≤k,

and (βi,j)0≤i≤n,0≤j≤k with the periodic property (2.47). Therefore, the following

theo-rem can be obtained.

Theorem 2.12 Given a basic admissible set _{B ⊂ Σ2×2}, an L-periodic pattern U is B-admissible if and only if

[β₀β_{1· · · βk−1}] and [β_k−1σn−l(β₀)] are _{B-admissible.} (2.49) Proposition 2.7 and Theorem 2.12 yield the following main results.

Theorem 2.13 For n_{≥ 1, 0 ≤ l ≤ n − 1 and k ≥ 1, denote by ΓB}     n l 0 k     the

cardinal number of the set of   n l

0 k 

-periodic and B-admissible patterns. For n ≥ 2, 0_{≤ l ≤ n − 1 and k ≥ 2,} Γ_B     n l 0 k     = tr Tk nRln
=_|Tk−1_{n ◦ R}n−l_n Tt_n| (2.50) and n−1 X l=0 Γ_B     n l 0 k     = tr Tk nRn
=_|Tk−1_{n ◦ R}nTtn|. (2.51)

For n≥ 2 and 0 ≤ l ≤ n − 1, Γ_B     n l 0 1     = tr(TnRnl) =|diag(Tn)◦ Rn−ln Ttn| (2.52) and n−1 X l=0 Γ_B     n l 0 1     = tr(TnRn) =|diag(Tn)◦ RnTtn|. (2.53) Furthermore, let T1 =   a1,1a1,1 a2,2a2,2 a3,3a3,3 a4,4a4,4   and R1 =   1 0 0 1   ; (2.54) then Γ_B     1 0 0 k     = tr(Tk 1). (2.55)

Proof. By Proposition 2.7, Theorem 2.12 and the construction of Tn, the results

(2.50) to (2.53) hold for n _{≥ 2, 0 ≤ l ≤ n − 1 and k ≥ 1.} For n = 1, define C_1×2= 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 , _(2.56)

which is the collection of x-periodic patterns of period 1 with height 2. Then, B-admissible patterns of C_1×2 are represented by T1 as defined in (2.54). Theorem 2.12

and the construction of T1 easily yields (2.55). The proof is complete.

The n-th order zeta function ζn(s) can now be obtained as follows.

Theorem 2.14 For n_{≥ 1,} ζn(s) = exp 1 n ∞ X k=1 1 ktr(T k nRn)skn ! . (2.57)

2.2 Rationality of ζn

This subsection proves that ζnis a rational function, as specified by (1.18). To elucidate

the method, the symmetric Tn is considered initially. For any n ≥ 1, let λj be an

eigenvalue of Tn: TnUj = λjUj, 1≤ j ≤ N and N ≡ 2n. If Tn is symmetric, then the

Jordan form of Tn [27] is

Tn= UJUt, (2.58)

where

Ut = U−1. (2.59)

The eigen-matrix U in (2.58) is defined by

U= [U1, U2,· · · , UN]N ×N = [ui,j]N ×N, (2.60)

where Uj = (u1,j, u2,j,· · · , uN,j)t is the j-th (column) eigenvector, and

J= diag(λ1, λ2,· · · , λN). (2.61)

Moreover, λj can be arranged such that λ1 ≥ |λ2| ≥ · · · ≥ |λN|. Equation (2.59)

implies

N

P

p=1

ui,puj,p = δi,j and N

P

q=1

uq,iuq,j = δi,j. (2.62)

Now, Theorem 2.15 will be proven.

Theorem 2.15 Assume Tn is symmetric; then

1 n n−1 X l=0 Γ_B     n l 0 k     = 1 ntr T k nRn
= X λ∈Σ(Tn) χ(λ)λk, (2.63)

where Σ(Tn) is the spectrum of Tn,

χ(λ) = X

λj=λ

and χ(λj) = _n1|Rn◦ UjUjt| = 1 n P i∈In ωn,i n _n−1 P l=0 uσl_(i),j 2 , (2.65)

where ωn,i is the cardinal number of Cn(i). Moreover,

ζn(s) = Y λ∈Σ(Tn) (1− λsn₎−χ(λ)_{. (2.66)} Proof. Clearly, tr Tk nRn
= tr (Udiag(λj)UtRn) = N P j=1 _N P i=1 ui,j N P p=1 up,j _n−1 P l=1 Rl n;p,i  λj. For each j, 1_{≤ j ≤ N,} N P i=1 ui,j  _N P p=1 up,j n−1_P l=0 Rl n;p,i  = PN i=1 ui,j _n−1 P l=0 uσ−l_(i),j  = P i∈In ωn,i n _n−1 P l=0 uσl_(i),j  _n−1 P l=0 uσ−l_(i),j  = P i∈In ωn,i n _n−1 P l=0 uσl_(i),j 2 .

The following is easily verified;

|Rn◦ UjUjt| = X i∈In ωn,i n n−1 X l=0 uσl_(i),j !2 . (2.67)

Then, (2.63)∼(2.65) follow. From [21],

∞ X k=1 1 kJ k_skn_{= diag log(1} − λjsn)−1
. (2.68)

Therefore, (2.66) holds. The proof is complete.

We now extend Theorem 2.15 to general Tn. In this case, the Jordan form for Tn

Tn = UJU−1, (2.69)

where U is given as (2.60) and Uj, 1 ≤ j ≤ N, is an eigenvector or generalized

eigenvector [21; 27]. Denote by

U−1 = [wi,j] = [W1; W2;· · · ; WN]_{N ×N} (2.70)

with Wi = (wi,1, wi,2,· · · , wi,N), the i-th row vector.

J= diag(J1, J2,· · · , JQ), (2.71)

where Jq is the Jordan block, 1≤ q ≤ Q:

Jq=           λq 1 0 · · · 0 0 0 λq 1 · · · 0 0 ... ... ... ... ... 0 0 0 · · · λq 1 0 0 0 _{· · ·} 0 λq           Mq×Mq , (2.72) Mq ≥ 1.

As is well-known [21], for any Jordan block

J =           λ 1 0 _{· · · 0 0} 0 λ 1 · · · 0 0 .. . ... ... ... ... 0 0 0 _{· · · λ 1} 0 0 0 · · · 0 λ           M ×M (2.73) and log(I − tJ) =           µ1,1 µ1,2 µ1,3 · · · µ1,M 0 µ2,2 µ2,3 · · · µ2,M . .. ... 0 0 · · · 0 µM,M           , (2.74) where

µ_i,i+j−1= µ1,j for 1≤ j ≤ M and 1 ≤ i ≤ M + 1 − j, (2.75) and µi,j = 0 if i > j. (2.76) In particular, 1_{≤ i ≤ M,} µi,i= log(1− λt). (2.77) Therefore, ∞ P k=1 1 kJ k_skn = _{− log(I − s}n_J) = _{−diag (log(I − s}n_J 1),· · · , log(I − snJQ)) = _−[µi,j]N ×N, (2.78) where log(I _{− s}nJq) =           µq;1,1 µq;1,2 µq;1,3 · · · µq;1,Mq 0 µq;2,2 µq;2,3 · · · µq;2,Mq . .. ... 0 0 · · · 0 µq;Mq,Mq           (2.79) and µq;i,i = log(1− λqsn), 1 ≤ q ≤ Q. (2.80)

Now, Theorem 2.15 is generalized for general Tn.

Lemma 2.16 For n≥ 1, in (2.69) and (2.70) the generalized eigen-matrix is denoted by

U= [U1,1· · · U1,M1;· · · ; Uq,1· · · Uq,Mq;· · · ; UQ,1· · · UQ,MQ]N ×N,

U−1 = [W1,1;· · · ; W1,M1;· · · ; Wq,1;· · · ; Wq,Mq;· · · ; WQ,1;· · · ; WQ,MQ]N ×N. Then, ζn(s) = Q Y q=1 Y 1≤i≤j≤Mq

exp (_−χq;i,jµq;i,j) , (2.81)

where χq;i,j = _n1|Rn◦ Uq,iWq,j| = 1 n P p∈In ωn,p n _n−1 P l=0 uq;σl_(p),i  _n−1 P l=0 wq;j,σl_(p)  . (2.82) In particular, if

χq;i,j = 0 for all i6= j, (2.83)

then ζn(s) = Q Q q=1 (1− λqsn)−χq = Q λ∈Σ(Tn) (1− λsn₎−χ(λ)_, (2.84) where χq = 1 n Mq X i=1 |Rn◦ Uq;iWq;i| (2.85) and χ(λ) = X λq=λ χq. (2.86)

Proof. From (2.69) and (2.78),

ζn(s) = exp  1 ntr U (−diag(log(I − s n_J 1),· · · , log(I − snJQ))) U−1Rn
 . Now,

tr (Udiag(log(I_{− s}n_J 1),· · · , log(I − snJQ))U−1Rn) = N P i=1 N P j=1 N P r=1 N P p=1 up,iµi,jwj,r _n−1 P l=0 Rl n;r,p  = N P i=1 N P j=1 P p∈In ωn,p n _n−1 P l=0 uσl_(p),i  _n−1 P l=0 wj,σ−l_(p)  µi,j.

Therefore, (2.81) follows. Clearly, if (2.83) holds, then (2.84) holds. The proof is complete.

In the rest of the section, (2.83) is proven and χ(λ) is shown to be a nonnegative integer. Therefore, ζn is a rational function. Some of the symmetry properties of the

eigenvectors associated with the Rn-symmetry of Tn are investigated first.

Lemma 2.17 For n_{≥ 1, if U is an eigenvector, then R}l

nU is also an eigenvector for

any 0 ≤ l ≤ n − 1. Furthermore, if U is a generalized eigenvector, then Rl

nU is also a

generalized eigenvector for any 0_{≤ l ≤ n − 1.}

Based on Lemma 2.17, the equivalent class _{R(U) of eigenvector U is introduced by} Rn.

Definition 2.18 For any N _{× 1 column vector U,} R(U) = Rl

nU|0 ≤ l ≤ n − 1

. (2.87)

U is called (Rn-) symmetric ifR(U) = {U}, such meaning that uj = uifor all j ∈ Cn(i)

or

Rl

nU = U (2.88)

for all 0_{≤ l ≤ n − 1. U is called (R}n-) anti-symmetric if n−1_P l=0 Rl nU = 0, such meaning n−1 X l=0 Uσl_(i) = 0 (2.89) for all i∈ In.

Lemma 2.19 Let U = (u1, u2,· · · , uN)t and W = (w1, w2,· · · , wN), 1 n|Rn◦ UW | = X i∈In 1 ωn,i   X j∈Cn(i) uj     X j∈Cn(i) wj   . (2.90) Furthermore, if U is symmetric, then

1 n|Rn◦ UW | = W U = N X j=1 ujwj, (2.91)

and if U is anti-symmetric, then 1 n|Rn◦ UW | = 0. (2.92) Proof. Clearly, n−1_P l=0 uσl_(i) = n ωn,i P j∈Cn(i) uj and n−1_P l=0 wσl_(i) = n ωn,i P j∈Cn(i) wj. (2.93)

Therefore, substituting (2.93) into (2.82) yields

1 n|Rn◦ UW | = X i∈In 1 ωn,i   X j∈Cn(i) uj     X j∈Cn(i) wj   . If U is symmetric, then X j∈Cn(i) uj = ωn,iui. Hence, 1 n|Rn◦ UW | = X i∈In   X j∈Cn(i) uiwj   = N X j=1 ujwj = W U.

The proof is complete.

The following orthogonal matrix Qn is very useful in finding symmetric and

anti-symmetric eigenvectors of Tn, the details of proof is omitted.

                       1 √ n 1 √ n 1 √ n · · · 1 √ n 1 √ n q n−1 n − 1 √ n(n−1) − 1 √ n(n−1) · · · − 1 √ n(n−1) − 1 √ n(n−1) 0 qn−2 n−1 − 1 √ (n−1)(n−2) · · · − 1 √ (n−1)(n−2) − 1 √ (n−1)(n−2) ... 0 0 0 · · · _√1 2 − 1 √ 2                        (2.94) is orthogonal.

In the following lemma, when Qnis used, R(U) can be expressed by symmetric and

anti-symmetric eigenvectors.

Lemma 2.21 For n≥ 2, given eigenvector U, define U1 = 1 √ n n−1 X l=0 Rl_nU (2.95) and, 2 ≤ j ≤ n, Uj = s n− j + 1 n_{− j + 2}R j−2 n U − 1 √ n_{− j + 1}√n_{− j + 2} n−1 X k=j−1 R_nkU. (2.96)

If _{R(U) has rank κ, for some κ, 1 ≤ κ ≤ n,} (i) then Uj

n

j=1 also has rank κ;

(ii) if U1 6= 0, then U1 is symmetric, and for each j, 2≤ j ≤ n, Uj is anti-symmetric.

Proof. Clearly, U1, U2,· · · , Un
t = Qn U, RnU,· · · , RjnU,· · · , Rn−1n U
t .

Since Qn is orthogonal, (i) holds.

n−1_P l=0 (Uj)σl_(i) = q n−j+1 n−j+2 n−1_P l=0 (Rj−2 n U)σl_(i)− 1 n−j+1 n−1_P k=j−1 n−1_P l=0 (Rk nU)σl_(i) ! =qn−j+1_n−j+2 n−1P l=0 uσl_(i)− 1 n−j+1 n−1_P k=j−1 n−1_P l=0 uσl_(i) ! = 0.

Therefore, Uj is anti-symmetric for any 2≤ j ≤ n. The proof is complete.

The main result can now be proven.

Theorem 2.22 For n≥ 1, 1 ntr T k nRn
= X λ∈Σ(Tn) χ(λ)λk (2.97) and ζn(s) = Y λ∈Σ(Tn) (1_{− λs}n)−χ(λ), (2.98) where χ(λ) is the number of linearly independent symmetric eigenvectors and general-ized eigenvectors of Tn with eigenvalue λ.

Proof. The case of symmetric Tn is considered first. Let Eλ be the eigenspace of Tn

with eigenvalue λ. By Lemma 2.21, Eλ is spanned by linearly independent symmetric

eigenvectors U1, U2,· · · , Up and anti-symmetric eigenvectors U1′, U2′,· · · , Up′′, where

p + p′ _{= dim(E}

λ) and p or p′ may be zero.

Now, χ(λ) = _n1 p P j=1|Rn◦ Uj Utj| + p′ P j=1|Rn◦ U ′ j(Uj′)t| ! = p, (2.99)

which is the number of linearly independent symmetric eigenvectors of Tn with

eigen-value λ.

For general Tn, in Jordan canonical form (2.69) and (2.71), U can be decomposed

into

Each Eλj is spanned by symmetric eigenvectors and generalized eigenvectors

Uj,1, Uj,2,· · · , Uj,pj and anti-symmetric eigenvectors and generalized eigenvectors

U′

j,1, Uj,2′ ,· · · , Uj,p′ ′

j, and pj + p

′

j = dim(Eλj).

The inverse matrix is U−1 ₌

h W1,1;· · · ; W1,p1; W1,1′ ;· · · ; W1,p′ ′ 1;· · · ; WQ,1;· · · ; WQ,pQ; W ′ Q,1;· · · ; WQ,p′ ′ Q i . Lemma 2.19 implies 1 n|Rn◦ Uj,iWj′,k| = δjj′δik and 1 n|Rn◦ Uj,i′ Wj′′_,k| = 0. Therefore, χ(λj) = pj

= the number of linearly independent symmetric eigenvectors and generalized eigenvectors of Tn with eigenvalue λj.

The result follows. The proof is complete.

Now, the reduced trace operator τn of Tn is recalled as in (1.17).

Definition 2.23 For n≥ 1, Tn = [tn;i,j]. For each i, j ∈ In, define

τn;i,j =

X

k∈Cn(j)

tn;i,k (2.100)

and denote the reduced trace operator of Tn by τn= [τn;i,j], which is a χn× χn matrix.

The following theorem indicates that τnis more effective in computing the

eigenval-ues with rotationally symmetric eigenvectors and generalized eigenvectors of Tn. See

also Examples 2.54 and 2.55.

Theorem 2.24 λ_{∈ Σ(T}n) with χ(λ) ≥ 1 if and only if λ ∈ Σ(τn). Moreover, χ(λ) is

the algebraic multiplicity of τn with eigenvalue λ. Furthermore,

1 n n−1 X l=0 Γ_B    n l 0 k     = X λ∈Σ(τn) χ(λ)λk= tr(τ_nk), (2.101) and ζn(s) = exp ∞ X k=1 tr(τk n) k s nk ! . (2.102)

Proof. Let λ ∈ Σ(Tn) be an eigenvalue with rotationally symmetric eigenvector

U = (u1, u2,· · · , u2n)t, where u_i = u_j for any i ∈ I_n and j ∈ C_n(i). Define V =

(u1,· · · , ui,· · · , u2n)t for i∈ I_n. Then, clearly, T_nU = λU implies τ_nV = λV .

On the other hand, if τnV = λV and V = (v1,· · · , vi,· · · , v2n)t, then V can be

extended to U, a 2n_{-vector, by u}

j = vi for i∈ Inand j ∈ Cn(i). Then, TnU = λU and

U is rotationally symmetric. The arguments also hold for a generalized eigenvector. Finally, (2.101) follows from (2.51) and (2.97), and (2.102) follows from (1.8) and (2.101). The proof is complete.

Remark 2.25 According to Theorem 2.24, the following is easily verified; X

λ∈Σ(Tn)

χ(λ) = X

λ∈Σ(τn)

χ(λ) = χn. (2.103)

Theorem 2.24 yields the following result.

Theorem 2.26 For n_{≥ 1,} ζn(s) = (det (I − snτn))−1 (2.104) = Y λ∈Σ(τn) (1_{− λs}n₎−χn(λ) , (2.105)

where χn(λ) is the algebraic multiplicity of λ ∈ Σ(τn) and

ζ(s) = ∞ Y n=1 (det (I− sn_τ n))−1 (2.106) = ∞ Y n=1 Y λ∈Σ(τn) (1_{− λs}n)−χn(λ) . (2.107)

2.3 More symbols on larger lattice

This subsection extends the results found in previous sections to any finite number of symbols

p_{≥ 2 on any finite square lattice Zm×m}, m_{≥ 2. The results are outlined here and the} details are left to the reader. The proofs of the theorems are omitted for brevity.

For fixed positive integers p≥ 2 and m ≥ 2, the set of symbols is denoted by Sp ={0, 1, 2, · · · , p − 1} and the basic square lattice is Zm×m. We need the following

For any fixed n ≥ m, such as in (2.14), the x-periodic patterns of period n with height m can be recorded as C_n×m;i,j in C_n×m by C_n×m;i,j =

β0,0 β0,0 β1,0 βn−1,0 β1,0 βm−2,0 β0,1 β0,1 β1,1 βn−1,1 β1,1 βm−2,1 β0,m−1 β0,m−1 β1,m−1 βn−1,m−1 β1,m−1 βm−2,m−1 Fig 2.1. .

Similarly, when 1_{≤ n ≤ m − 1, Cn×m}= [C_n×m;i,j] can also be defined as an (n + m_{− 1) × m pattern in Fig 2.1.}

Then, for any n≥ 1, the associated trace operator Tn×m= [tn×m;i,j] can be defined

by

t_n×m;i,j = 1 if and only if C_n×m;i,j isB-admissible. (2.108) Now, for any n_{≥ 1, the corresponding rotational matrix Rn×(m−1)} which is a zero-one pn(m−1)_{× p}n(m−1) _{matrix is defined by}

R_{n×(m−1);i,j} = 1 if and only if

j = σ(i), (2.109)

where i is given by 1_{≤ i ≤ p}n(m−1) _{and 1≤ σ(i) ≤ p}n(m−1) _{is represented by}

σ(i) = ψ [σ(β₀)σ(β₁)_{· · · σ(βm−2})]
. (2.110) The explicit expression for R_n×(m−1), like (2.31), can also be obtained and the result is omitted here.

C_n×(m−1)(i) = _{σj_{(i)|0 ≤ j ≤ n − 1}} =  jRl n×(m−1) 

i,j = 1 for some 1≤ l ≤ n

 ,

(2.111)

and the index set _In×(m−1) of n is defined by

In×(m−1)= {i|1 ≤ i ≤ pn(m−1), i≤ σq(i), 1 ≤ q ≤ n − 1} = i1 ≤ i ≤ pn(m−1)_{, i≤ j for all j ∈ C} n×(m−1)(i) . (2.112)

The cardinal number of In×(m−1) is denoted by χn×(m−1) and χn×(m−1) is equal to the

number of necklaces that can be made from 2m−1 _{colors, when the necklaces can be}

rotated but not turned over [48]. χ_n×(m−1) is expressed as

χ_n×(m−1) = 1 n

X

d|n

φ(d) 2m−1
n/d_{. (2.113)}

Like Proposition 2.3, R_n×(m−1) has the permutation properties. Now, define

R_n×(m−1)=

n−1

X

l=0

R_n×(m−1)l . (2.114)

A similar result to Theorem 2.13 can now be obtained for Γ_B     n l 0 k    . Theorem 2.27 For n_{≥ 1, k ≥ 1 and 0 ≤ l ≤ n − 1,}

Γ_B     n l 0 k     = tr Tk n×mRn×(m−1)l
(2.115) and n−1 X l=0 Γ_B     n l 0 k     = tr Tk n×mRn×(m−1)
. (2.116)

As in (1.8), the n-th order zeta function is given by

ζn(s) = exp  1 n ∞ X k=1 n−1 X l=0 1 kΓB     n l 0 k     skn   . (2.117) From Theorem 2.27, the following theorem is obtained.

Theorem 2.28 For any n≥ 1, ζn(s) = exp 1 n ∞ X k=1 1 ktr T k n×mRn×(m−1)
snk ! . (2.118)

The proof that ζn(s) is a rational function depends on the fact that Tn×m is also

R_n×(m−1)-symmetric, which is stated as follows.

Proposition 2.29 For any n_{≥ 1,}

t_{n×m;σ(i),σ(j)}= t_n×m;i,j (2.119)

for any 1≤ i, j ≤ pn(m−1)_.

Then the reduced trace operator τ_n×m of T_n×m is defined as follows.

Definition 2.30 For n_{≥ 1, the reduced trace operator τn×m} = [τ_n×m;i,j] of T_n×m is a χ_n×(m−1)× χn×(m−1) matrix defined by

τ_n×m;i,j = X

k∈Cn×(m−1)(j)

t_n×m;i,k (2.120)

for each i, j ∈ In×(m−1).

The notion of symmetric and anti-symmetric eigenvectors of T_n×m can also be defined as in Definition 2.18. Now, the main result can be obtained.

Theorem 2.31 For any n≥ 1,

ζn(s) =

Y

λ∈Σ(Tn×m)

(1− λsn₎−χ(λ) _(2.121)

= (det (I_{− s}nτ_n×m))−1, (2.122) where χ(λ) is the number of linearly independent symmetric eigenvectors and general-ized eigenvectors of T_n×m with eigenvalue λ. The zeta function is

ζ(s) = ∞ Y n=1 (det (I− sn_τ n×m))−1. (2.123)