Multiple Dedekind Zeta Functions

TL;DR

This paper introduces multiple Dedekind zeta values (MDZV) as a generalization of Euler's multiple zeta values using iterated integrals on a membrane, explores their properties, and presents conjectures based on explicit residue computations.

Contribution

It defines MDZV via a new integral approach, relates them to multiple Eisenstein series, and studies their analytic properties and special values.

Findings

MDZV can be expressed as infinite sums and have integral representations.

Explicit computation of a multiple residue at (1,1,1,1) over quadratic fields.

Formulation of two conjectures about MDZV based on residue analysis.

Abstract

In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series (Gangl, Kaneko and Zagier). We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein-Kronecker series (Goncharov). Each of them is a special case of multiple Dedekind zeta values. MDZV are interpolated into functions that we call multiple Dedekind zeta functions (MDZF). We show that MDZF have integral representation, can be written as infinite sum, and have analytic continuation. We compute explicitly the value of a multiple residue of certain MDZF over a quadratic number field at the point (1,1,1,1). Based on such computations, we state two conjectures about MDZV.