Finite canonical partition function at the bosonic Hagedorn threshold ($Z_1({\beta }_H)<\infty \text{but}{Z}_1\text{ is nonanalytic at }{\beta }_H$)

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 ${\beta }_H=4\pi \sqrt{{\alpha }^{\prime }}$. 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.