  The first result of our article is another proof of Mertens' third theorem in
the number field setting, which generalises a method of Hardy. The second
result concerns the sign of the error term in Mertens' third theorem. Diamond
and Pintz showed that the error term in the classical case changes sign
infinitely often and in our article, we establish this result for number fields
assuming a reasonable technical condition. In order to do so, we needed to
prove Cram\'er's inequality for number fields, which is interesting in its own
right. Lamzouri built upon Diamond and Pintz's work to prove the existence of
the logarithmic density of the set of real numbers $x \ge 2$ such that the
error term in Mertens' third theorem is positive, so the third result of our
article generalises Lamzouri's results for number fields. We also include
numerical investigations for the number fields $\mathbb{Q}(\sqrt{5})$ and
$\mathbb{Q}(\sqrt{13})$, building upon similar work done by Rubinstein and
Sarnak in the classical case.
