The class number formula conjecture for the Carlitz module

Let $k[t]$ be the polynomial ring underlying the Carlitz module $E$, let $R$ be the ring from the preceding construction, and let $H_R$ be the cokernel of the map $E(R)\to \mathrm{H}^1(G,\Lambda_E)$. Define

$$\zeta_R(1)=\sum_{I\subset R}\frac{1}{|R/I|},$$

where $I$ ranges over the nonzero ideals of $R$, and let ${\rm Reg}_R$ be the regulator defined as the unique monic representative of the determinant of the natural map from the period lattice to ${\rm Lie}_E(R)$.

The class number formula conjecture.

$$\zeta_R(1)={\rm Reg}_R\cdot |H_R|.$$

This is the conjectural analogue of the classical class number formula for the Carlitz module. In the source, the conjecture is stated to reduce to the identity $\zeta_R(1)=\log(1)$, which was proven by Carlitz; accordingly, the conjecture is recorded as solved.