Related papers: Partial zeta functions of algebraic varieties over…
The knowledge on irrationality of p-adic zeta values has recently progressed. The irrationality of zeta_2(2), \zeta_2(3) and of a few other p-adic series of Dirichlet was obtained by F. Calegari. F. Beukers gave a more elementary proof of…
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,…
We extend some definitions and give new results about the theory of slice analysis in several quaternionic variables. The sets of slice functions which are respectively slice, slice regular and circular w.r.t. given variables are…
In this paper we continue to study the Reidemeister zeta function. We prove P\'olya -- Carlson dichotomy between rationality and a natural boundary for analytic behavior of the Reidemeister zeta function for a large class of automorphisms…
The theory of Ihara zeta functions is extended to infinite graphs which are weighted and of finite total weight. In this case one gets meromorphic instead of rational functions and the classical determinant formulas of Bass and Ihara hold…
The higher rank Lefschetz formula for p-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a p-adic group on its Bruhat-Tits building. By specializing to certain lines one gets…
We introduce and study multivariate zeta functions enumerating subrepresentations of integral quiver representations. For nilpotent such representations defined over number fields, we exhibit a homogeneity condition that we prove to be…
This theorem is based on holomorphy of studied functions and the fact that near a singularity point the real part of some rational function can take an arbitrary preassigned value.
In [P] R. Pellikaan introduced a two variable zeta-function for a curve over a finite field and proved that it is a rational function. Here we show that its denominator is absolutely irreducible. This is motivated by work of J. Lagarias and…
This is the first of four papers that study algebraic and analytic structures associated to the Lerch zeta function. This paper studies "zeta integrals" associated to the Lerch zeta function using test functions, and obtains functional…
It is known that not all summation methods are linear and stable. Zeta function regularization is in general non-linear. However, in some cases formal manipulations with "zeta function" regularization (assuming linearity of sums) lead to…
In this paper we introduce new generalizations of the zeta function, the Tricomi functions; their main properties are studied. This opens the way to a deeper, better application of these functions both in the theory of special functions,…
We define the zeta function of a finite category. And we propose a conjecture which states the relationship between the Euler characteristic of finite categories and the zeta function of finite categories. This conjecture is verified when…
This note describes continued fraction representations for the rational approximations to the zeta function recently found by the author. It is tempting to think that these continued fractions might be analysed using a souped up version of…
By studying the spectral aspects of the fractional part function in a well-known separable Hilbert space, we show, among other things, a rational approximation of the Riemann zeta function and its derivatives valid on every vertical line in…
The (partially) ordered set of the non-trivial zeros of the zeta function with positive imaginary parts is considered. The order is the coordinatewise order inherited from $\mathbb{C}$. Some interesting properties regarding the minimal…
We study some infinite products of absolute zeta functions. Especially, we consider the convergence and the rationality of them.
We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta…
This note contains a short proof of the functional equation for the zeta function.
The topological zeta function of a matroid is a rational function as well as a valuative invariant of the matroid, encoding rich combinatorial information. We analyze topological zeta functions of matroids from the vantage point of several…