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

TL;DR

This paper extends Euler's theorem on the divergence of the sum of reciprocals of primes to the setting of algebraic number fields, showing a similar divergence for certain sums over principal ideals in the ring of integers.

Contribution

It generalizes Euler's classical result to number fields, linking divergence of series over integers to divergence over principal ideals in algebraic number theory.

Findings

Divergence of sum over integers implies divergence over associated principal ideals.

Generalization of Euler's theorem to algebraic number fields.

Establishes a connection between series over integers and ideals in number fields.

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.