Secondary terms in the distribution of genus numbers of cubic fields

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.

arXiv commentCorrected coefficients in Theorem 2.1 and the derived results (including the main theorems). Also simplified the arguments by removing the unnecessary twisting by quadratic characters