= Finite canonical partition function at the bosonic Hagedorn threshold
{title2=$Z_1(\beta_H)<\infty\quad\text{but}\quad Z_1\text{ is nonanalytic at }\beta_H$}
For a critical <closed bosonic string> in 25 noncompact spatial dimensions, the smoothed <string oscillator> rest-mass density behaves as $\rho(M)\sim C M^{-26}e^{\beta_H M}$, with $\beta_H=4\pi\sqrt{\alpha'}$. At large $M$, the centre-of-mass <momentum> integral of $e^{-\beta\sqrt{M^2+p^2}}$ is asymptotic to $(2\pi M/\beta)^{25/2}e^{-\beta M}$, up to the convention for phase-space measure. Thus the high-mass tail of the one-string <canonical partition function> is proportional to $\int_{M_0}^\infty M^{-27/2}e^{-(\beta-\beta_H)M}\,dM$. This converges at $\beta=\beta_H$ but fails for $\beta<\beta_H$. Expanding about the endpoint gives twelve finite derivatives and a nonanalytic term proportional to $(\beta-\beta_H)^{25/2}$; the thirteenth derivative diverges. Consequently exponential state growth does not imply divergence of the <partition function> at the threshold itself. The discussion assumes the separate infrared instability of the bosonic <tachyon> has been regulated and does not claim a stable bosonic thermal vacuum.
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