  In arXiv:1706:09426 we conjectured and provided evidence for an identity
between Siegel $\Theta$-constants for special Riemann surfaces of genus $n$ and
products of Jacobi $\theta$-functions. This arises by comparing two different
ways of computing the \nth \Renyi entropy of free fermions at finite
temperature. Here we show that for $n=2$ the identity is a consequence of an
old result due to Fay for doubly branched Riemann surfaces. For $n>2$ we
provide a detailed matching of certain zeros on both sides of the identity.
This amounts to an elementary proof of the identity for $n=2$, while for $n\ge
3$ it gives new evidence for it. We explain why the existence of additional
zeros renders the general proof difficult.
