Unconditional Explicit Mertens' Theorems for Number Fields and Dedekind Zeta Residue Bounds

TL;DR

This paper establishes explicit, unconditional bounds for Mertens' theorems in number fields, providing effective constants and error bounds based on the degree and discriminant, advancing the understanding of Dedekind zeta residues.

Contribution

It introduces unconditional, effective versions of Mertens' theorems for number fields with explicit constants and error bounds tied to field invariants.

Findings

Explicit bounds for Dedekind zeta residue at s=1

Effective constants for Mertens' theorems in number fields

Error terms explicitly depend on degree and discriminant

Abstract

We obtain unconditional, effective number-field analogues of the three Mertens' theorems, all with explicit constants and valid for $x\geq 2$. Our error terms are explicitly bounded in terms of the degree and discriminant of the number field. To this end, we provide unconditional bounds, with explicit constants, for the residue of the corresponding Dedekind zeta function at $s=1$.