Related papers: Dynamical Zeta Functions in the Nonorientable Case
We study the magnetic geodesic flows on 2-surfaces having an additional first integral which is independent of the Hamiltonian at a fixed energy level. The following two cases are considered: when there exists a quadratic in momenta…
Many examples of zeta functions in number theory and combinatorics are special cases of a construction in homotopy theory known as a decomposition space. This article aims to introduce number theorists to the relevant concepts in homotopy…
We show some examples of topological zeta functions associated to an isolated plane curve singular point and an allowed, in the sense of N\'emethi and Veys, differential form that have several poles of order two. This is in contrast to the…
In this paper we study spectral zeta functions associated to finite and infinite graphs. First we establish a meromorphic continuation of these functions under some general conditions. Then we study special values in the case of standard…
In this paper, we study the twisted Ruelle zeta function associated with the geodesic flow of a compact, hyperbolic, odd-dimensional manifold $X$. The twisted Ruelle zeta function is associated with an acyclic representation $\chi\colon…
The objective of the paper is twofold. The first objective is to study the uniqueness problem of meromorphic function $f(z)$ when $f^{(1)}(z)$ shares two distinct finite values $a_1$, $a_2$ and $\infty$ CM with $\Delta_cf(z)$. In this…
Periodic orbit theory provides two important functions---the dynamical zeta function and the spectral determinant for the calculation of dynamical averages in a nonlinear system. Their cycle expansions converge rapidly when the system is…
We present a method for computing the zeta function of a smooth projective variety over a finite field which proceeds by induction on the dimension. We have implemented our approach for some surfaces using the Magma programming language,…
We construct geometrically infinite hyperbolic surfaces supporting horocycles with tailored recurrence properties. In particular, we obtain the first examples of non-trivial minimal horocyclic orbit closures and of infinite locally-finite…
In various contexts, the zeta function of an object splits into a product of $L$-functions. We categorify this product formula for quadratic covers of objects in the following contexts: quadratic extensions of number fields, ramified double…
We describe the topology of superlevel sets of ($\alpha$-stable) L\'evy processes X by introducing so-called stochastic $\zeta$-functions, which are defined in terms of the widely used $\text{Pers}_p$-functional in the theory of persistence…
We give explicit formulas for the Betti numbers, both stable and unstable, of the unordered configuration spaces of an arbitrary surface of finite type.
We describe a construction of random meromorphic functions with prescribed simple poles with unit residues at a given stationary point process. We characterize those stationary processes with finite second moment for which, after…
We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. As an illustration, we explain how…
We develop techniques for computing zeta functions associated with nilpotent groups, not necessarily associative algebras, and modules, as well as Igusa-type zeta functions. At the heart of our method lies an explicit convex-geometric…
We study fixed points of iterates of dynamically affine maps (a generalisation of Latt\`es maps) over algebraically closed fields of positive characteristic $p$. We present and study certain hypotheses that imply a dichotomy for the…
The meromorphic function $W(s)$ introduced in the Riemann-Zeta function $\zeta(s) = W(s) \zeta(1-s)$ maps the line of $s = 1/2 + it$ onto the unit circle in $W$-space. $|W(s)| = 0$ gives the trivial zeroes of the Riemann-Zeta function…
We prove a mixed joint discrete universality theorem for a Matsumoto zeta-function $\varphi(s)$ (belonging to the Steuding subclass) and a periodic Hurwitz zeta-function $\zeta(s,\alpha;{\mathfrak{B}})$. For this purpose, certain…
To test a possible relation between the topological entropy and the Arnold complexity, and to provide a non trivial example of a rational dynamical zeta function, we introduce a two-parameter family of two-dimensional discrete rational…
For a finite group $G$, we consider the zeta function $\zeta_G(s) = \sum_{H} \abs{H}^{-s}$, where $H$ runs over the subgroups of $G$. First we give simple examples of abelian $p$-group $G$ and non-abelian $p$-group $G'$ of order $p^m, \; m…