Multiple Dedekind zeta functions and distribution of prime numbers represented by polynomials

Contact: Pieter Moree (moree@mpim...)

Multiple Dedekind zeta functions are generalization of multiple zeta functions to any number field K. If we fix the field K, we consider primes p=N(x+y\alpha) represented as a norm of an element of K,  x and y being integers The distribution of the quotient y/x is given by an analogue of a multiple Dedekind zeta value at s=2, (where the variable is the cone of integration). Experimentally it is tested for many quadratic fields, some cubic and a few of degree 4,5 or 6. For cubic fields, the integrant that gives the distribution of y/x is a rational function on an elliptic curve. We have formulated and tested conjectures for distribution of pairs of prime numbers p=N(gamma), q=N(gamma+delta), where gamma and delta are from the same cone. The distribution is given by a multiple Dedekind zeta value at (2,2). For example, for the Gaussian integers, the multiple Dedekind zeta value describes the actual distribution of pairs of primes with error less than 0.1%.