An asymptotic on the logarithms of the relative class numbers of imaginary abelian number fields of prime conductor and large degree

An asymptotic on the logarithms of the relative class numbers of the cyclotomic number fields of prime conductors $p$ is known. Here we give an asymptotic on the logarithms of the relative class numbers of the imaginary abelian number fields of prime conductors $p$ and large degrees $m=(p-1)\mathrm{/}d$ with $\phi (d)=o(\log p)$. We also show that this asymptotic does not hold true anymore under the only slightly weaker restriction $\phi (d)=O(\log p)$.