Secondary terms in the distribution of genus numbers of cubic fields

TL;DR

This paper establishes the existence of secondary terms in the asymptotic formulas for the average and distribution of genus numbers of cubic fields, improving error estimates and providing uniform moment estimates.

Contribution

It introduces the first proof of secondary terms of order X^{5/6} in the distribution of genus numbers of cubic fields, refining previous estimates.

Findings

Confirmed secondary terms of order X^{5/6} in asymptotic formulas.

Provided uniform estimates for moments of genus numbers.

Improved error bounds over previous results.

Abstract

We prove the existence of secondary terms of order $X^{5/6}$ in the asymptotic formulas for the average size of the genus number of cubic fields and for the number of cubic fields with a given genus number, establishing improved error estimates. These results refine the estimates obtained by McGown and Tucker. We also provide uniform estimates for the moments of the genus numbers of cubic fields.