On a two-variable zeta function for number fields

Recently van der Geer and Schoof [11, Prop. 1] formulated an "exact" analogue of the Riemann-Roch theorem for an algebraic number field K, based on Arakelov divisors. They used this result to formally express the completed zeta function ζ_K(s) of K as an integral over the Arakelov divisor class group Pic(K) of K. They introduced a two-variable zeta function attached to a number field K, also given as an integral over the Arkelov class group, which we call either the Arakelov zeta function or the two-variable zeta function. This zeta function was modelled after a two-variable zeta function attached to a function field over a finite filed, introduced in 1996 by Pellikaan [18]. For convenience we review the Arakelov divisor interpretation of the two-variable zeta function and the Riemann-Roch theorem for number fields in an appendix.