Mertens' Third Theorem for Number Fields: A New Proof, Cramér's Inequality, Oscillations, and Bias

Abstract

The first result of our article is another proof of Mertens' third theorem in the generalised setting of number fields, which generalises a method of Hardy. Diamond and Pintz showed that the error term in the classic Mertens' third theorem changes sign infinitely often, so the second result of our article generalises Diamond and Pintz's result for number fields. That is, subject to some conditions, we show that the error term in Mertens' third theorem for number fields changes sign infinitely often. To prove this result, we also needed to prove Cramér'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 x 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.