Related papers: Functional equations for Weng's zeta functions for…

Associated to classical semi-simple groups and their maximal parabolics are genuine zeta functions. Naturally related to Riemann's zeta and governed by symmetries, including that of Weyl, these zetas are expected to satisfy the Riemann…

We prove certain general forms of functional relations among Witten multiple zeta-functions in several variables (or zeta-functions of root systems). The structural background of those functional relations is given by the symmetry with…

Let $L$ be a solvable Lie algebra of dimension less than or equal to 4 over finite fields. We compute and record, in explicit symbolic form, the zeta functions enumerating subalgebras or ideals of $L$, and study their properties. We also…

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 a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional…

We define zeta-functions of weight lattices of compact connected semisimple Lie groups. If the group is simply-connected, these zeta-functions coincide with ordinary zeta-functions of root systems of associated Lie algebras. In this paper…

We introduce new non-abelian zeta functions for curves defined over finite fields. There are two types, i.e., pure non-abelian zetas defined using semi-stable bundles, and group zetas defined for pairs consisting of (reductive group,…

By restricting the variables running over various (possibly different) subfields, we introduce the notion of a partial zeta function. We prove that the partial zeta function is rational in an interesting case, generalizing Dwork's well…

We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local representation zeta functions associated with…

We prove that all but finitely many zeros of Weng's zeta function for a Chevalley group defined over $\Q$ are simple and on the critical line under some reasonable geometric hypothesis.

We write down the functional equation of the zeta function of a global field. This equation is implicit in Weil's ``Basic Number Theory''.

We study zeta-functions of weight lattices of compact connected semisimple Lie groups of type $A_3$. Actually we consider zeta-functions of SU(4), SO(6) and PU(4), and give some functional relations and new classes of evaluation formulas…

The exceptional group $G_2$ has two maximal parabolic subgroups $P_{long}$, $P_{short}$ corresponding to the so-called long root and short root. In this paper, the second author introduces two zeta functions associated to $(G_2,P_{long})$…

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…

We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on…

We consider zeta functions with values in the Grothendieck ring of Chow motives. Investigating the lambda-structure of this ring, we deduce a functional equation for the zeta function of abelian varieties. Furthermore, we show that the…

We study zeta functions enumerating subalgebras or ideals of Lie algebras over finite field of prime order $\mathbb{F}_p$. We first develop a general blueprint method for computing zeta functions of $\mathbb{F}_p$-Lie algebras, and…

We provide the formula of motivic zeta function for semi-quasihomogeneous singularities and in dimension two, we determine the poles of zeta functions. We also give another formula for stringy E-function using embedded…

In this article we define and study a zeta function $\zeta_G$ - similar to the Hasse-Weil zeta function - which enumerates absolutely irreducible representations over finite fields of a (profinite) group $G$. The zeta function converges on…

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…