Related papers: Coh zeta functions for inert quadratic orders
The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading…
The motivic zeta function of a smooth and proper $\mathbb{C}((t))$-variety $X$ with trivial canonical bundle is a rational function with coefficients in an appropriate Grothendieck ring of complex varieties, which measures how $X$…
We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups $O(n)$. We describe the critical loci of the quadratic trace function Tr$(AXBX^T)$ and determine their indices via perfect…
We consider the prehomogeneous vector space of pairs of ternary quadratic forms. For the lattice of pairs of integral ternary quadratic forms and its dual lattice, there are six zeta functions associated with the the prehomogeneous vector…
We investigate classification results for general quadratic functions on torsion abelian groups. Unlike the previously studied situations, general quadratic functions are allowed to be inhomogeneous or degenerate. We study the discriminant…
In his work studying the Zeta functions of families of hypersurfaces, Dwork came upon a one-parameter family of hypersurfaces (now known as \emph{the} Dwork family). These examples were not only useful to Dwork in his study of his…
The Argand diagram is used to display some characteristics of the Riemann Zeta function. The zeros of the Zeta function on the complex plane give rise to an infinite sequence of closed loops, all passing through the origin of the diagram.…
We address a class of definite integrals known as Berndt-type integrals, highlighting their role as specialized instances within the integral representation framework of the Barnes-zeta function. Building upon the foundational insights of…
The multiplicative structure of the trivial symplectic groupoid over $\mathbb R^d$ associated to the zero Poisson structure can be expressed in terms of a generating function. We address the problem of deforming such a generating function…
In this paper, we establish a new zeta function based on the Bartholdi zeta function for an undirected graph G called the reduced Bartholdi zeta function. We study the relation between its coefficients and the structure of the graph, and…
We obtain a family of strict $\hat G$-invariant products on the space of holomorphic functions on a semisimple coadjoint orbit of a complex connected semisimple Lie group $\hat G$. By restriction, we also obtain strict $G$-invariant…
A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structure, plus countably many special…
The cotangent zeta function is a very interesting object, which is related to partial zeta functions and Hecke $L$-functions of real quadratic fields. Its special values at odd integers greater than 1 are explicitly evaluated by Berndt in…
We discuss about the conjectural cohomological theory of dynamical zeta functions in the case of general Anosov flows. Our aim is to provide a functional-analytic framework that enables us to justify the basic part of the theory rigorously.…
A family of multiple integrals over four variables is rewritten in terms of a family of simple integrals involving the product of four modified Bessel (Macdonald functions). The latter are shown to be related to 7zeta(3)/2. A generalization…
For the Tits building B(G) of a finite group of Lie type G(Fq), we study the edge zeta function, which enumerates edge-geodesic cycles in the 1-skeleton. We show that every nonzero edge eigenvalue becomes a power of q after raising to a…
We propose a generalization of the classical theta function to higher cohomology of the polarization line bundle on a family of complex tori with positive index. The constructed cocycles vary horizontally with respect to the (projective)…
In this paper we define and study a Dedekind-like zeta function for the algebra of multicomplex numbers. By using the idempotent representations for such numbers, we are able to identify this zeta function with the one associated to a…
We survey some recent applications of p-adic cohomology to machine computation of zeta functions of algebraic varieties over finite fields of small characteristic, and suggest some new avenues for further exploration.
We study ``forms of the Fermat equation'' over an arbitrary field $k$, i.e. homogenous equations of degree $m$ in $n$ unknowns that can be transformed into the Fermat equation $X_1^m+...+X_n^m$ by a suitable linear change of variables over…