Double shuffle relations for multiple Dedekind zeta values

Ivan Horozov¹

¹ Department of Mathematics and Computer Science City University of New York Bronx Community College 2155 University Avenue Bronx, NY 10453, U.S.A.

Acta Arithmetica, Tome 180 (2017) no. 3, pp. 201-227

Résumé

We define two types of shuffle relations for multiple Dedekind zeta values over an imaginary quadratic field. Multiple Dedekind zeta values were defined by the author (2014). We give examples of integral shuffle relations in terms of iterated integrals over membranes, and of infinite sum shuffle relations. Then we establish both types of shuffles in general, which represent a product of two ordinary multiple Dedekind zeta values as a sum of such in two ways. This leads to a linear relation among ordinary multiple Dedekind zeta values for imaginary quadratic fields. We also present explicit formulas for the two types of shuffle for the product $\zeta _{K}(2)\zeta _{K}(2)$.

Détail

DOI : 10.4064/aa7945-8-2016

Keywords: define types shuffle relations multiple dedekind zeta values imaginary quadratic field multiple dedekind zeta values defined author examples integral shuffle relations terms iterated integrals membranes infinite sum shuffle relations establish types shuffles general which represent product ordinary multiple dedekind zeta values sum ways leads linear relation among ordinary multiple dedekind zeta values imaginary quadratic fields present explicit formulas types shuffle product zeta zeta