Related papers: A trace on fractal graphs and the Ihara zeta funct…

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$ is the graph whose vertices correspond to the $i(G)$-sets, and where two…

For a given infinite connected graph $G=(V,E)$ and an arbitrary but fixed conductance function $c$, we study an associated graph Laplacian $\Delta_{c}$; it is a generalized difference operator where the differences are measured across the…

We intimate deeper connections between the Riemann zeta and gamma functions than often reported and further derive a new formula for expressing the value of $\zeta(2n+1)$ in terms of zeta at other fractional points. This paper also…

In this article we review the observation, due originally to Dwork, that the zeta-function of an arithmetic variety, defined originally over the field with p elements, is a superdeterminant. We review this observation in the context of a…

The zeta-dimension of a set A of positive integers is the infimum s such that the sum of the reciprocals of the s-th powers of the elements of A is finite. Zeta-dimension serves as a fractal dimension on the positive integers that extends…

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta(s)$, and hence Riemann's function $\zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(i\kappa)$ of…

We consider the alternating zeta function and the alternating $L$-function of a graph $G$, and express them by using the Ihara zeta function of $G$. Next, we define a generalized alternating zeta function of a graph, and express the…

The paper deals with the analytic entire function Chi(s) closely related to Riemann Zeta Function Zeta(s). A formula is obtained for Chi(s) essentially within the so-called critical strip. This is achieved by applying Cauchy integral…

We study analytic properties of the representation zeta functions of arithmetic groups of type $\mathsf{A}_2$, such as $\textrm{SL}_3(\mathbb{Z})$. In particular, we uncover further poles of these functions and determine a natural boundary…

In this paper we obtain the set of grafts from some set of 12 elementary functions on 12 critical strips. We make use this directly for grafting of elements of the corresponding set of $\zeta$-factorization formulas. This procedure gives…

We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of `relative compactness' for such graphs and study sufficient and necessary conditions for…

In this paper we extend the Zeta function regularization technique, which gives a meaningful solution to divergent power series, in order to assign finite values to divergent integral of certain transcendental functions $f(x)$. The…

The zeta function attached to a finite complex $X_\Gamma$ arising from the Bruhat-Tits building for $\PGL_3(F)$ was studied in \cite{KL}, where a closed form expression was obtained by a combinatorial argument. This identity can be…

We study zeta functions enumerating submodules invariant under a given endomorphism of a finitely generated module over the ring of ($S$-)integers of a number field. In particular, we compute explicit formulae involving Dedekind zeta…

We introduce multifractal pressure and dynamical multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the fine multifractal spectra of graph-directed…

We show that certain representations of graphs by operators on Hilbert space have uses in signal processing and in symbolic dynamics. Our main result is that graphs built on automata have fractal characteristics. We make this precise with…

Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space ${\mathbb R}^N$, for any integer $N\ge1$. It is defined by…

The zeta function of an integral lattice $\Lambda$ is the generating function $\zeta_{\Lambda}(s) = \sum\limits_{n=0}^{\infty} a_n n^{-s}$, whose coefficients count the number of left ideals of $\Lambda$ of index $n$. We derive a formula…

In finite graphs, finite-order tangles offer an abstract description of highly connected substructures. In infinite graphs, infinite-order tangles compactify the graphs in the same way the ends compactify connected locally finite graphs.…

Goss zeta values can be found, in some cases, as evaluations of a new type of rigid analytic function on projective curves $X$ over a finite field $\mathbb{F}_q$, called "Pellarin $L$-series". In the case of genus $0$ and $1$, Pellarin and…