Related papers: Ruelle zeta function for cofinite hyperbolic Riema…

This paper studies the non-holomorphic Eisenstein series E(z,s) for the modular surface, and shows that integration with respect to certain non-negative measures gives meromorphic functions of s that have all their zeros on the critical…

As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple…

We introduce new zeta functions related to an endomorphism $\phi$ of a discrete group $\Gamma$. They are of two types: counting numbers of fixed ($\rho\sim \rho\circ\phi^n$) irreducible representations for iterations of $\phi$ from an…

The Stieltjes constants gamma_k(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about its only pole at s=1. We present the relation of gamma_k(1) to the eta_j coefficients that appear in the Laurent…

We consider a Riemmaniann compact manifold $M$, the associated Laplacian $\Delta$ and the corresponding Brownian motion $X_t$, $t\geq 0.$ Given a Lipschitz function $V:M\to\mathbb R$ we consider the operator $\frac{1}{2}\Delta+V$, which…

For any $a\in\mathbb{C}$, the zeros of $\zeta(s)-a$, denoted by $\rho_a=\beta_a+i\gamma_a$, are called $a$-points of the Riemann zeta function $\zeta(s)$. In this paper, we reformulate some basic results about the $a$-points of $\zeta(s)$…

The present paper is devoted to finding a necessary and sufficient condition on the occurence of scattering for the regularly hyperbolic systems with time-dependent coefficients whose time-derivatives are integrable over the real line. More…

We show that the Ruelle zeta function of any smooth Axiom A flow with orientable stable/unstable spaces has a meromorphic continuation to the entire complex plane. The proof uses the meromorphic continuation result of [arXiv:1410.5516]…

We study the microlocal properties of the scattering matrix associated to the semiclassical Schr\"odinger operator $P=h^2\Delta_X+V$ on a Riemannian manifold with an infinite cylindrical end. The scattering matrix at $E=1$ is a linear…

We consider solutions to some semilinear elliptic equations on complete noncompact Riemannian manifolds and study their classification as well as the effect of their presence on the underlying manifold. When the Ricci curvature is…

Moduli spaces of stable parabolic bundles of parabolic degree $0$ over the Riemann sphere are stratified according to the Harder--Narasimhan filtration of underlying vector bundles. Over a Zariski open subset $\mathscr{N}_{0}$ of the open…

We study the nonlinear Helmholtz equation $(\Delta - \lambda^2)u = \pm |u|^{p-1}u$ on $\mathbb{R}^n$, $\lambda > 0$, $p \in \mathbb{N}$ odd, and more generally $(\Delta_g + V - \lambda^2)u = N[u]$, where $\Delta_g$ is the (positive)…

This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by $\exp(C_K |s|^{\delta})$ in strips $|\Re s| \leq K$, where…

Fried's conjecture is concerned with the behavior of dynamical zeta functions at the origin. For compact hyperbolic manifolds, Fried proved that for an orthogonal acyclic representation of the fundamental group, the twisted Ruelle zeta…

We initiate the study of spectral zeta functions $\zeta_{X}$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions.…

The secondary zeta function $Z(s)=\sum_{n=1}^\infty\alpha_n^{-s}$, where $\rho_n=\frac12+i\alpha_n$ are the zeros of zeta with $\Im(\rho)>0$, extends to a meromorphic function on the hole complex plane. If we assume the Riemann hypothesis…

We consider the Steklov zeta function $\zeta$ $\Omega$ of a smooth bounded simply connected planar domain $\Omega$ $\subset$ R 2 of perimeter 2$\pi$. We provide a first variation formula for $\zeta$ $\Omega$ under a smooth deformation of…

Let $O$ be a one-dimensional Cohen-Macaulay local ring having a finite field as a coefficient field. The aim of this work is to extend the explicit computations of the St\"ohr Zeta Function of $O$ for one and two branches to an arbitrary…

The Riemann hypothesis, stating that the real part of all non-trivial zero points fo the zeta function must be $\frac{1}{2}$, is one of the most important unproven hypothesises in number theory. In this paper we will proof the Riemann…

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the…