Multiple Dedekind zeta functions and evaluations of extended multiple zeta values

We define the number field analog of the zeta function of d-complex variables studied by Zagier in (First European Congress of Mathematics, vol. II (Paris, 1992), Progress in Mathematics, vol. 120, Birkhauser, Basel, 1994, pp. 497-512). We prove that in certain cases this function has a meromorphic continuation to C^d, and we identify the linear subvarieties comprising its singularities. We use our approach to meromorphic continuation to prove that there exist infinitely many values of these functions at regular points in their extended domains which can be expressed as a rational linear combination of values of the Dedekind zeta function.     

Multiple q-Mahler measures and zeta functions

We introduce multiple q-Mahler measures and we calculate some specific examples, where multiple q-analogues of zeta functions appear. We study also limits as the multiple q goes to 1.     

Partial zeta functions

In the present paper, we study analytic properties of the zeta functions defined by the Euler products over elements in subsets of the set of prime elements.     

On invariant measures of extensions of Markov transition functions

By using the LITTLEWOOD matrices A_2n we generalize CLARKSON' S inequalities, or equivalently, we determine the norms ‖A_2n: l(L_P) → l(L_P)‖ completely. The result is compared with the norms ‖A_2n: l → l‖, which are calculated implicitly in PIETSCH [6].     

Shintani-Barnes zeta and gamma functions

We show that Shintani's work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations. We re-prove many of Shintani's formulas and prove several new ones. Among the latter is a generalization to the Shintani-Barnes gamma functions of Raabe's 1843 formula , and a further generalization to the Shintani zeta functions. These explicit formulas can be interpreted as “vanishing period integral” side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions. We also relate Barnes’ triple gamma function to the elliptic gamma function appearing in connection with certain integrable systems.     

Fractional hypergeometric zeta functions

In this paper we investigate a continuous version of the hypergeometric zeta functions for any positive rational number “a” and demonstrate the analytic continuation. The fractional hypergeometric zeta functions are shown to exhibit many properties analogous to its hypergeometric counter part, including its intimate connection to Bernoulli numbers.

     

p-adic zeta functions and Bernoulli numbers

One gives a new proof to the Leopoldt-Kubota-Iwasawa theorem regarding the possibility of the p-adic interpolation of the values of the Riemann zeta-function and of the Dirichlet L-functions at negative integral points. To this end, for each root ? ≠ 1 of unity one introduces and one investigates the numbers C_n(?) which arise in the expansion $$\frac{{\varepsilon - 1}}{{\varepsilon e^z - 1}} = \sum\limits_{n = 0}^\infty {\frac{{C_n (\varepsilon )}}{{n!}}Z^n }$$ One proves a generalization of the Kummer congruences for the Bernoulli numbers.     

On zeta functions and Iwasawa modules

We study the relation between zeta-functions and Iwasawa modules. We prove that the Iwasawa modules for almost all determine the zeta function when is a totally real field. Conversely, we prove that the -part of the Iwasawa module is determined by its zeta-function up to pseudo-isomorphism for any number field Moreover, we prove that arithmetically equivalent CM fields have also the same -invariant.

     

Series involving zeta and related functions

Using Padé approximation to the exponential function, we obtain new identities involving values of the Rieman zeta function at integers. Applications to series associated with zeta numbers are proved. In particular, expansion of ζ(3) (resp. ζ(5)) in terms of ζ(4j + 2) (resp. ζ(4j)) are proved.     

Rank metric codes and zeta functions

We define the rank metric zeta function of a code as a generating function of its normalized q-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank metric codes. We further prove a functional equation and derive an upper bound for the minimum distance in terms of the reciprocal roots of the zeta function. Finally, we show invariance under suitable puncturing and shortening operators and study the distribution of zeroes of the zeta function for a family of codes.     

Rationality of partial zeta functions

We prove that the partial zeta function introduced in [9] is a rational function, generalizing Dwork's rationality theorem.     

Edge zeta functions of hypergraphs

We introduce the edge zeta function of a hypergraph and present its determinant expression. Furthermore, we give a decomposition formula for the edge zeta function of a group covering of a hypergraph. Finally, we present two new determinant expressions for the edge zeta function of a hypergraph.     

Ihara zeta functions of digraphs

After defining and exploring some of the properties of Ihara zeta functions of digraphs, we improve upon Kotani and Sunada’s bounds on the poles of Ihara zeta functions of undirected graphs by considering digraphs whose adjacency matrices are directed edge matrices.     

Bartholdi zeta functions of digraphs

We give a determinant expression for the Bartholdi zeta function of a digraph which is not symmetric. This is a generalization of Bartholdi’s result on the Bartholdi zeta function of a graph or a symmetric digraph.