scholarly journals Bartholdi Zeta Functions for Hypergraphs

10.37236/1003 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Iwao SATO
Keyword(s):  
Zeta Function ◽  
Bipartite Graph ◽  
Zeta Functions ◽  
Selberg Zeta Function ◽  
Decomposition Formula

Recently, Storm defined the Ihara-Selberg zeta function of a hypergraph, and gave two determinant expressions of it. We define the Bartholdi zeta function of a hypergraph, and present a determinant expression of it. Furthermore, we give a determinant expression for the Bartholdi zeta function of semiregular bipartite graph. As a corollary, we obtain a decomposition formula for the Bartholdi zeta function of some regular hypergraph.

scholarly journals Second variation of Selberg zeta functions and curvature asymptotics

2019 ◽  
Vol 57 (1) ◽  
pp. 23-60
Author(s):  
Ksenia Fedosova ◽  
Julie Rowlett ◽  
Genkai Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Vector Bundle ◽  
Zeta Function ◽  
Explicit Formula ◽  
Zeta Functions ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Second Variation ◽  
Selberg Zeta Function ◽  
The Second Variation

Abstract We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, Z(s), on Teichmüller space. We then use this formula to determine the asymptotic behavior as $$\mathfrak {R}s \rightarrow \infty $$Rs→∞ of the second variation. As a consequence, for $$m \in {\mathbb {N}}$$m∈N, we obtain the complete expansion in m of the curvature of the vector bundle $$H^0(X_t, {\mathcal {K}}_t)\rightarrow t\in {\mathcal {T}}$$H0(Xt,Kt)→t∈T of holomorphic m-differentials over the Teichmüller space $${\mathcal {T}}$$T, for m large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $$O(m^2 \mathrm{e}^{-l_0 m}),$$O(m2e-l0m), where $$l_0$$l0 is the length of the shortest closed hyperbolic geodesic.

scholarly journals Weighted Zeta Functions of Graph Coverings

10.37236/1117 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Iwao Sato
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Equivalence Classes ◽  
Signed Graphs ◽  
Base Graph ◽  
Regular Covering ◽  
Decomposition Formula ◽  
Graph Coverings

We present a decomposition formula for the weighted zeta function of an irregular covering of a graph by its weighted $L$-functions. Moreover, we give a factorization of the weighted zeta function of an (irregular or regular) covering of a graph by equivalence classes of prime, reduced cycles of the base graph. As an application, we discuss the structure of balanced coverings of signed graphs.

scholarly journals Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems

1996 ◽  
Vol 16 (4) ◽  
pp. 805-819 ◽  
Author(s):  
Hans Henrik Rugh
Keyword(s):  
Dynamical Systems ◽  
Zeta Function ◽  
Complex Plane ◽  
Entire Functions ◽  
Fredholm Determinant ◽  
Three Dimensional ◽  
Zeta Functions ◽  
Selberg Zeta Function ◽  
Fredholm Determinants ◽  
Axiom A

AbstractWe consider a generalized Fredholm determinant d(z) and a generalized Selberg zeta function ζ(ω)−1 for Axiom A diffeomorphisms of a surface and Axiom A flows on three-dimensional manifolds, respectively. We show that d(z) and ζ(ω)−1 extend to entire functions in the complex plane. That the functions are entire and not only meromorphic is proved by a new method, identifying removable singularities by a change of Markov partitions.

scholarly journals THE IHARA–SELBERG ZETA FUNCTION FOR PGL3 AND HECKE OPERATORS

2006 ◽  
Vol 17 (02) ◽  
pp. 143-155 ◽  
Author(s):  
ANTON DEITMAR ◽  
J. WILLIAM HOFFMAN
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Hecke Operators ◽  
Weak Version ◽  
Selberg Zeta Function ◽  
Tits Building ◽  
Arithmetical Interpretation

A weak version of the Ihara formula is proved for zeta functions attached to quotients of the Bruhat–Tits building of PGL3. This formula expresses the zeta function in terms of Hecke-operators. It is the first step towards an arithmetical interpretation of the combinatorially defined zeta function.

Zeta functions of twisted modular curves

2006 ◽  
Vol 80 (1) ◽  
pp. 89-103 ◽  
Author(s):  
Cristian Virdol
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Modular Curves ◽  
Absolute Galois Group ◽  
Modular Curve ◽  
The Absolute

AbstractIn this paper we compute and continue meromorphically to the whole complex plane the zeta function for twisted modular curves. The twist of the modular curve is done by a modprepresentation of the absolute Galois group.

scholarly journals MOTIVIC HILBERT ZETA FUNCTIONS OF CURVES ARE RATIONAL

2018 ◽  
Vol 19 (3) ◽  
pp. 947-964
Author(s):  
Dori Bejleri ◽  
Dhruv Ranganathan ◽  
Ravi Vakil
Keyword(s):  
Zeta Function ◽  
Generating Function ◽  
Zeta Functions ◽  
Hilbert Schemes ◽  
Grothendieck Ring

The motivic Hilbert zeta function of a variety $X$ is the generating function for classes in the Grothendieck ring of varieties of Hilbert schemes of points on $X$. In this paper, the motivic Hilbert zeta function of a reduced curve is shown to be rational.

scholarly journals A New Determinant Expression for the Weighted Bartholdi Zeta Function of a Digraph

10.37236/2649 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Iwao Sato ◽  
Hideo Mitsuhasi ◽  
Hideaki Morita
Keyword(s):  
Zeta Function ◽  
Zeta Functions

We consider the weighted Bartholdi zeta function of a digraph $D$, and give a new determinant expression of it. Furthermore, we treat a weighted $L$-function of $D$, and give a new determinant expression of it. As a corollary, we present determinant expressions for the Bartholdi edge zeta functions of a graph and a digraph.

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