Related papers: Zeta Functions of the Dirac Operator on Quantum Gr…

We generalize earlier studies on the Laplacian for a bounded open domain $\Omega\in \real^2$ with connected complement and piecewise smooth boundary. We compare it with the quantum mechanical scattering operator for the exterior of this…

In this paper, we study the Dirichlet series that enumerates proper equivalence classes of full-rank sublattices of a given quadratic lattice in a hyperbolic plane -- that is, a nondegenerate isotropic quadratic space of dimension $2$. We…

We study zero modes of Laplacians on compact and non-compact metric graphs with general self-adjoint vertex conditions. In the first part of the paper the number of zero modes is expressed in terms of the trace of a unitary matrix…

We study the probability that a cycle of length k in the lattice [1, n]^s does not contain more lattice points than the k vertices of the cycle. Then we generalize this problem to other configurations induced by a given graph H,…

We introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends Kedlaya's algorithm to a very general class of curves using a map to the projective line. We develop all the necessary bounds,…

We define a new weighted zeta function for a finite graph and obtain its determinant expression. This result gives the characteristic polynomial of the transition matrix of the Szegedy walk on a graph.

In this survey we gather recent results on Dirac operators coupled with $\delta$-shell interactions. We start by discussing recent advances regarding the question of self-adjointness for these operators. Afterward we switch to an…

The aim of the present paper is to analyse the spectrum of Laplace and Dirac type operators on metric graphs. In particular, we show for equilateral graphs how the spectrum (up to exceptional eigenvalues) can be described by a natural…

In this paper we define a symmetric zeta function. We show that it can be analytically continued to a meromorphic function on $\mathbb{C}^3$ with only simple poles at some special hyperplanes. We also calculate the value of a multiple…

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 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…

The complexity of a graph can be obtained as a derivative of a variation of the zeta function or a partial derivative of its generalized characteristic polynomial evaluated at a point [\textit{J. Combin. Theory Ser. B}, 74 (1998), pp.…

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to…

Given an open set $\Omega\subset\mathbb{R}^3$. We deal with the spectral study of Dirac operators of the form $H_{a,\tau}=H+A_{a,\tau}\delta_{\partial\Omega}$, where $H$ is the free Dirac operator in $\mathbb{R}^3$, $A_{a,\tau}$ is a…

We determine the zeta functions of trinomial curves in terms of Gauss sums and Jacobi sums, and we obtain an explicit formula of the genus of a trinomial curve over a finite field, then we study the conditions for a trinomial curve to be a…

An analysis of the zeta and gamma function is presented, using elementary functions like [] and {}, a general formula for the angle of zeta(1/2 + i*n) is found and the same for the gamma function.

The half-line Dirac operators with $L^2$-potentials can be characterized by their spectral data. It is known that the spectral correspondence is a homeomorphism: close potentials give rise to close spectral data and vice versa. We prove the…

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 introduce the multiple zeta functions with structures similar to those of symmetric functions such as Schur $P$-, Schur $Q$-, symplectic and orthogonal functions in the representation theory. We first consider their basic properties such…

Let P be a selfadjoint elliptic operator of order m>0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at…