Oscillations in Mertens’ product theorem for number fields

The content of this talk is based on joint work with Shehzad Hathi. First, I will give a short but sweet proof of Mertens’ product theorem for number fields, which generalises a method introduced by Hardy. Next, when the number field is the rationals, we know that the error in this result changes sign infinitely often. Therefore, a natural question to consider is whether this is always the
case for any number field? I will answer this question (and more) during the talk. Furthermore, I will present the outcome of some computations in two number fields: $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\sqrt{13})$.