Large-level degeneracy of an open bosonic string ($d_p\sim 2^{-1/2}{p}^{-27/4}{e}^{4\pi \sqrt{p}}$)

The Dedekind eta function modular transformation gives $F(e^{-t})\sim (t/2\pi {)}^{12}\exp (4{\pi }^2/t-t)$. The saddle-point approximation for coefficient extraction has $t_0=2\pi /\sqrt{p-1}$, phase $4\pi \sqrt{p-1}$ and curvature $(p-1{)}^{3/2}/\pi$. Its prefactor $t_0^{12}$ supplies $(p-1{)}^{-6}$ and the Gaussian width supplies $(p-1{)}^{-3/4}/\sqrt{2}$. This explains both the exponential growth and the power $-27/4$.