Mertens' theorem for Chebotarev sets

TL;DR

This paper extends Mertens' theorem to Chebotarev sets of primes in Galois extensions, providing new formulas and computations for various types of number field extensions and prime representations.

Contribution

It generalizes Mertens' product theorem to Chebotarev sets in Galois extensions, broadening its applicability and including explicit computations for specific cases.

Findings

Extended Mertens' theorem to Chebotarev sets in Galois extensions

Computed products for Chebotarev sets in abelian, S_3 sextic, and quadratic form primes

Demonstrated the theorem's applicability to various number field extensions

Abstract

We generalize Mertens' product theorem to Chebotarev sets of prime ideals in Galois extensions of number fields. Using work of Rosen, we extend an argument of Williams from cyclotomic extensions to this more general case. Additionally, we compute these products for Cheboratev sets in abelian extensions, $S_3$ sextic extensions, and sets of primes represented by some quadratic forms.