= Large-level degeneracy of a closed bosonic string
{title2=$g_N\sim\tfrac12(N-1)^{-27/2}e^{8\pi\sqrt{N-1}}$}
The left and right transverse oscillator sectors each have <generating function> $\prod_{m\ge1}(1-q^m)^{-24}$. Its coefficient satisfies $d_N\sim2^{-1/2}(N-1)^{-27/4}e^{4\pi\sqrt{N-1}}$. <Closed-string level matching> requires equal levels but leaves their polarizations independent, so $g_N=d_N^2$. With $\alpha'M^2=4(N-1)$, the exponential is $e^{\beta_H M}$ with $\beta_H=4\pi\sqrt{\alpha'}$. The smooth rest-mass density is proportional to $M^{-26}e^{\beta_H M}$, since $dN/dM=\alpha'M/2$. This distinguishes degeneracy per level from density per unit mass. The <polynomial> prefactor matters at the boundary of canonical convergence.
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