= Large-level degeneracy of an open bosonic string
{title2=$d_p\sim2^{-1/2}p^{-27/4}e^{4\pi\sqrt p}$}
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 $(p-1)^{-3/4}/\sqrt2$. This explains both the exponential growth and the power $-27/4$.
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