On Sums Over Prime Ideals

In 2017 Speyer proved a rational theorem for certain sums over irreducible polynomials of the polynomial ring \(A = \mathbb {F}_q[\theta ]\) $A={\mathbb{F}}_q[\theta ]$ . In this paper we generalize Speyer’s theorem for sums over prime ideals for an arbitrary coefficient ring A. Our approach is based on the theory of Goss’s zeta values and that of Drinfeld–Hayes A-modules related to the class field theory.