Mertens's theorem for splitting primes and more

TL;DR

This paper revisits Mertens's theorem, highlighting Hardy’s elegant proof using Landau's Tauberian theorem, and extends the theorem to primes in arithmetic progressions and non-abelian cases.

Contribution

It introduces a new proof approach for Mertens's theorem and extends its applicability to primes in arithmetic progressions and non-abelian groups.

Findings

Hardy's proof provides a direct and elegant derivation of Mertens's theorem.

The theorem is extended to primes in arithmetic progressions with uniformity in the modulus.

A non-abelian analogue of Mertens's theorem is established.

Abstract

Myriad articles are devoted to Mertens's theorem. In yet another, we merely wish to draw attention to a proof by Hardy, which uses a Tauberian theorem of Landau that "leads to the conclusion in a direct and elegant manner". Hardy's proof is also quite adaptable, and it is readily combined with well-known results from prime number theory. We demonstrate this by proving a version of the theorem for primes in arithmetic progressions with uniformity in the modulus, as well as a non-abelian analogue of this.