Secondary Term of Asymptotic Distribution of $S_3\times A$ Extensions over $\mathbb{Q}$

TL;DR

This paper establishes a refined asymptotic count with a secondary term for $S_3 imes A$ extensions over $Q$, using sieve methods and uniformity estimates, and demonstrates a power-saving error term for such groups.

Contribution

It introduces a novel combination of sieve techniques and uniformity estimates to improve the asymptotic counting of $S_3 imes A$ extensions over $Q$, including secondary terms and error bounds.

Findings

Proved a secondary term in the asymptotic count of $S_3 imes A$ extensions.

Established the existence of a power-saving error term for these extensions.

Extended results to odd abelian groups with minimal prime divisor greater than 5.

Abstract

We combine a sieve method together with good uniformity estimates to prove a secondary term for the asymptotic estimate of $S_3\times A$ extensions over $\mathbb{Q}$ when $A$ is an odd abelian group with minimal prime divisor greater than $5$. At the same time, we prove the existence of a power saving error when $A$ is any odd abelian group.