The average genus number for pure fields of prime degree

Let $\mathrm{\ell }\ge 5$ be prime. Let ${\mathcal{F}}_{\mathrm{\ell }}$ be the collection of (isomorphism classes of) pure number fields $\mathbb{Q}(\sqrt[\mathrm{\ell }]{a})$ of degree $\mathrm{\ell }$, ordered by the absolute value of their discriminant. In 2018, Benli proved a counting theorem for ${\mathcal{F}}_{\mathrm{\ell }}$, generalizing a previous theorem of Cohen and Morra when $\mathrm{\ell }=3$. We prove that the proportion of pure fields of degree $\mathrm{\ell }$ with genus number one is asymptotic to $(A_{\mathrm{\ell }}\log X{)}^{-1}$ and that the average genus number for pure fields of degree $\mathrm{\ell }$ is asymptotic to $B_{\mathrm{\ell }}(\log X{)}^{\mathrm{\ell }-1}$. Both $A_{\mathrm{\ell }}$ and $B_{\mathrm{\ell }}$ are expressed explicitly as a product over primes.