On Selberg's Lemma For Algebraic Fields

Abstract

1. Introduction. Recently two Japanese authors (1) gave a beautifully simple proof of Selberg's fundamental lemma in the theory of distribution of primes. The proof is based on a curious twist in the Möbius inversion formula. The object of this note is to show that their proof may be extended to a proof of the result for algebraic fields corresponding to Selberg's lemma. Shapiro (2) has already derived this result using Selberg's methods and deduced as a consequence the prime ideal theorem.