= Solution
Use units $k_B=\hbar=c=1$. In critical <light-cone gauge in string theory>, each chiral sector of the <closed bosonic string> has 24 transverse <string oscillator> species. Their <generating function> is
$$
F(q)=\prod_{m\geq1}(1-q^m)^{-24}=\sum_{N\geq0}d_Nq^N.
$$
Each factor sums the independent occupation numbers of the <string oscillators> at level $m$. For $q=e^{-t}$ with $t\downarrow0$, the modular transformation of the <Dedekind eta function> gives
$$
F(e^{-t})\sim\left(\frac{t}{2\pi}\right)^{12}\exp\left(\frac{4\pi^2}{t}-t\right).
$$
Extract the coefficient by a contour integral and apply the <saddle-point approximation> to the exponent $(N-1)t+4\pi^2/t$. The saddle is $t_*=2\pi/\sqrt{N-1}$, with exponent $4\pi\sqrt{N-1}$ and second derivative $(N-1)^{3/2}/\pi$. The factor $t^{12}$ gives $(N-1)^{-6}$, and the Gaussian width gives $(N-1)^{-3/4}/\sqrt2$. Hence
$$
d_N\sim\frac1{\sqrt2}(N-1)^{-27/4}e^{4\pi\sqrt{N-1}}.
$$
For a noncompact <closed string>, <closed-string level matching> imposes equal left and right levels, but does not correlate their transverse polarizations. Therefore the <large-level degeneracy of a closed bosonic string> is
$$
\boxed{g_N=d_N^2\sim\frac12(N-1)^{-27/2}e^{8\pi\sqrt{N-1}}.}
$$
The mass relation is $\alpha'M^2=4(N-1)$. Thus $8\pi\sqrt{N-1}=\beta_HM$ with
$$
\boxed{\beta_H=4\pi\sqrt{\alpha'},\qquad T_H=\frac1{4\pi\sqrt{\alpha'}}.}
$$
Distinguish degeneracy per discrete level from a smoothed density per unit rest mass. Since $dN/dM=\alpha'M/2$, the latter is
$$
\rho(M)\sim C M^{-26}e^{\beta_HM}.
$$
The exponential growth comes from the many ways of distributing <string oscillator> energy along a string; it is much faster than a fixed finite collection of particle species. The corresponding high-energy <entropy> is $S(E)=\beta_HE+O(\log E)$, so its microcanonical <temperature> tends to $T_H$. Additional energy can go into very long strings rather than raising the <temperature> substantially.
The Boltzmann suppression competes with this exponential density. For the <string oscillator> rest-mass sum alone the large-$M$ tail is $\int dM\,M^{-26}e^{-(\beta-\beta_H)M}$. It converges for $\beta>\beta_H$ and fails for $\beta<\beta_H$. To include translational states in 25 noncompact spatial dimensions, take the one-string <partition function> per unit spatial volume. At large $M$,
$$
\int d^{25}p\,e^{-\beta\sqrt{M^2+p^2}}\sim\left(\frac{2\pi M}{\beta}\right)^{25/2}e^{-\beta M}.
$$
This follows by expanding the energy as $M+p^2/(2M)+\cdots$ and doing a Gaussian <momentum> integral. Its high-mass contribution is consequently proportional to
$$
\int_{M_0}^\infty dM\,M^{-27/2}e^{-(\beta-\beta_H)M}.
$$
This integral is finite at $\beta=\beta_H$, although nonanalytic there: its first twelve derivatives in $\beta$ are finite, while the thirteenth diverges. The endpoint expansion contains a term proportional to $(\beta-\beta_H)^{25/2}$. Thus one must not assert that the <canonical partition function> necessarily diverges exactly at $T_H$; the polynomial prefactor and the spatial spectrum matter. This is the <finite canonical partition function at the bosonic Hagedorn threshold> distinction. The multi-string ideal gas has the same leading Hagedorn threshold, since the higher occupation-cycle terms probe $\ell\beta$, $\ell\geq2$, and are nonsingular at $\beta_H$.
There is also a <spacetime> diagnosis. Euclidean <temperature> makes time a circle of radius $R_\beta=\beta/(2\pi)$. A <string oscillator> ground state winding once around that circle, with zero <momentum> number, has
$$
m_{\rm th}^2=\frac{R_\beta^2}{\alpha'^2}-\frac4{\alpha'}=\frac{\beta^2}{4\pi^2\alpha'^2}-\frac4{\alpha'}.
$$
It becomes massless at $\beta_H$ and tachyonic for $T>T_H$. This <thermal winding instability of a closed bosonic string> signals failure of the perturbative low-temperature gas and motivates a transition to a different high-energy phase, often described in terms of long-string or winding-mode condensation. Free state counting fixes the threshold and its singular behavior, not the interacting phase's endpoint or the order of a transition.
Finally, the ordinary bosonic <tachyon> is already an infrared instability at zero <temperature>. The above canonical discussion is a formal analysis of the high-level spectrum with that separate infrared problem regulated. It does not establish a stable thermal vacuum of the unmodified bosonic theory; an actual equilibrium transition requires an appropriate stable setting and control of interactions.
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