  We compute a Riemann-Roch formula for the invariant Riemann-Roch number of a
quantizable Hamiltonian $S^1$-manifold $(M,\omega,\mathcal{J})$ in terms of the
geometry of its symplectic quotient, allowing $0$ to be a singular value of the
moment map $\mathcal{J}:M\to\mathbb{R}$. The formula involves a new explicit
local invariant of the singularities. Our approach relies on a complete
singular stationary phase expansion of the associated Witten integral.
