Kummer's relative class number conjecture

Let $q$ be a prime, let $h_1(q)$ be the relative class number of the cyclotomic field $\mathbb Q(\zeta_q)$ over its maximal real subfield, and define

$$G(q)=2q\left(\frac{q}{4\pi^2}\right)^{(q-1)/4},\qquad R(q)=\frac{h_1(q)}{G(q)}.$$

The Kummer conjecture. As $q$ tends to infinity, $R(q)$ tends to $1$. This is the classical asymptotic prediction for relative class numbers; the paper later gives average bounds, but the conjecture itself is not resolved here.