On the conical zeta values and the Dedekind zeta values for totally real fields

TL;DR

This paper establishes a relationship between Dedekind zeta values for totally real fields and conical zeta values, expressing partial zeta functions as rational combinations of conical zeta values linked to algebraic cones, scaled by the square root of the discriminant.

Contribution

It introduces a novel connection between Dedekind zeta functions and conical zeta values for totally real fields, expanding the understanding of their algebraic and analytic properties.

Findings

Partial zeta functions expressed as rational linear combinations of conical zeta values.

Relation holds up to the square root of the discriminant of the field.

Provides explicit formulas linking algebraic cones to zeta values.

Abstract

The conical zeta values are a generalization of the multiple zeta values which are defined by certain multiple sums over convex cones. In this paper, we present a relation between the values of the Dedekind zeta functions for totally real fields and the conical zeta values for certain algebraic cones. More precisely, we show that the values of the partial zeta functions for totally real fields can be expressed as a rational linear combination of the conical zeta values associated with certain algebraic cones up to the square root of the discriminant.