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

TL;DR

This paper presents a new proof of Mertens' third theorem for number fields, investigates the sign oscillations of the error term, proves Cramér's inequality in this context, and includes numerical analysis for specific quadratic fields.

Contribution

It introduces a novel proof of Mertens' third theorem for number fields, establishes sign oscillation results, and proves Cramér's inequality in this setting, extending classical results.

Findings

New proof of Mertens' third theorem for number fields

Sign oscillations of the error term are established for number fields

Numerical investigations for quadratic fields $\

Abstract

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.