A generalization to number fields of Euler's theorem on the series of reciprocals of primes

Abstract

Let $X$ be a set of positive integers, and let ${\mathbb{Z}}_K$ be the ring of integers of a number field $K$ of degree $n$. Denote by $N(I)$ the absolute norm of an ideal $I$ of ${\mathbb{Z}}_K$, and by $\mathcal{A}$ the set of principal ideals $a{\mathbb{Z}}_K$ such that $a$ is an atom of ${\mathbb{Z}}_K$ and $a$ divides $m$ for some $m\in X$. Building upon the ideas of Clarkson from [Proc. Amer. Math. Soc. 17 (1966), 541], we show that, if the series ${\sum }_{m\in X}1/m$ diverges, then so does the series ${\sum }_{\mathfrak{a}\in \mathcal{A}}|N(\mathfrak{a}){|}^{-1/n}$. Most notably, this generalizes a classical theorem of Euler on the series of reciprocals of positive rational primes.

4 pages, no figures. Refined the main theorem and added a few closing remarks. Comments are welcome