  In 1997, Y. Ohno empirically stumbled on an astoundingly simple identity
relating the number of cubic rings $h(D)$ of a given discriminant $D$, over the
integers, to the number of cubic rings $h'(D)$ of discriminant $-27D$ in which
every element has trace divisible by 3: $h'(D) = h(D)$, if $D < 0$, and $h'(D)
= 3h(D)$ if $D > 0$, where in each case, rings are weighted by the reciprocal
of their number of automorphisms. This allows the functional equations
governing the analytic continuation of the Shintani zeta functions (the
Dirichlet series built from the values of $h$ and $h'$) to be put in
self-reflective form. In 1998, J. Nakagawa verified Ohno's conjecture. We
present a new proof of the identity that uses the main ingredients of
Nakagawa's proof (binary cubic forms, recursions, and class field theory), as
well as one of Bhargava's celebrated higher composition laws, while aiming to
stay true to the stark elegance of the identity.
