Secondary terms in the distribution of genus numbers of cubic fields

We prove the existence of secondary terms of order $X^{5\mathrm{/}6}$ 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.