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.