Multiple Dedekind Zeta Functions
Seminar Room 1, Newton Institute
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 them as generalizations of Euler's multiple zeta values to arbitrary number fields. Over imaginary quadratic fields MDZV capture in particular multiple Eisenstein series [ZGK]. We give an analogue of multiple Eisenstein series over real quadratic field. And an alternative definition of values of multiple Eisenstein-Kronecker series [G]. Each of them as 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 and can be written as infinite sums, and have an analytic continuation. Finally, we prove that the multiple residue of a multiple Dedekind zeta function at (1,...,1) is a period.