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Use units $k_B=\hslash =c=1$. 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 isEach factor sums the independent occupation numbers of the string oscillators at level $m$. For $q=e^{-t}$ with $t\downarrow 0$, the modular transformation of the Dedekind eta function givesExtract 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 $t_{\ast }=2\pi /\sqrt{N-1}$, 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 $(N-1{)}^{-3/4}/\sqrt{2}$. HenceFor 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 isThe mass relation is ${\alpha }^{\prime }{M}^2=4(N-1)$. Thus $8\pi \sqrt{N-1}={\beta }_{H}M$ withDistinguish degeneracy per discrete level from a smoothed density per unit rest mass. Since $dN/dM={\alpha }^{\prime }M/2$, the latter isThe 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 $S(E)={\beta }_{H}E+O(\log E)$, 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 $M$,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 toThis 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 $\ell \beta$, $\ell \ge 2$, 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, hasIt 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.