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Start by using the metric equation of the Polyakov action to retain the Virasoro constraints after choosing conformal gauge. In the light-cone gauge in string theory, make $X^{+}$ proportional to worldsheet time on a patch with nonzero $k^{+}$. The constraints determine $X^{-}$ from the transverse coordinates, so its oscillators are not independent physical excitations. Neumann boundary conditions make each transverse coordinate a cosine standing wave on the open interval. Quantize these waves as independent harmonic oscillators and build their Fock space on the momentum-labelled vacuum. There are $d-2=24$ transverse directions in the critical bosonic string, and the normal-ordering intercept is one. Thus physical oscillator states have nonnegative occupation numbers $r_{n,i}$ and level $p={\sum }_{n,i}n{r}_{n,i}$, with $M^2=(p-1)/{\alpha }^{\prime }$. Continuous center-of-mass momentum is not part of the discrete oscillator degeneracy.

For each mode $n$ and transverse direction, summing its occupation numbers gives ${\sum }_{r\ge 0}{w}^{nr}=(1-w^n{)}^{-1}$. Therefore the open-string oscillator degeneracy generating function isThis is the Cauchy coefficient formula. The original PDF's power $w^{-p}$ is an index error: it extracts $d_{p-1}$, and would give zero for the vacuum $p=0$, which actually has one oscillator state. Any positively oriented circle strictly inside the unit disk works.

Put $w=e^{-t}$ with $t>0$. Expand the logarithm and sum the geometric series to obtainSince $t/(e^{mt}-1)\le 1/m$, dominated convergence gives $t\log F(e^{-t})\to 24{\sum }_{m\ge 1}{m}^{-2}=4{\pi }^2$. Thus the leading singular term is $4{\pi }^2/t$.

The subleading term printed in the PDF also needs correction: the product implies $+12\log ϵ$, rather than $-6\log ϵ$. To obtain the coefficient needed for the final power law, use $\tau =it/(2\pi )$ and the Dedekind eta function. Its definition gives $F(e^{-t})=e^{-t}\eta (\tau {)}^{-24}$. The modular identity $\eta (-1/\tau )=\sqrt{-i\tau }\eta (\tau )$, recorded in NIST's eta transformation formulas, and the product at large imaginary argument giveSince $t=-\log (1-ϵ)=ϵ+{ϵ}^2/2+O({ϵ}^3)$, this yieldsThe printed logarithmic correction cannot be used to derive the stated $p^{-27/4}$ prefactor; the corrected correction does.

For the saddle-point approximation, change the coefficient contour to $t=-\log w$. Its local contribution isSet $q=p-1$. The positive saddle is $t_0=2\pi /\sqrt{q}$, with $S(t_0)=4\pi \sqrt{q}$ and $S^{\prime \prime }(t_0)=q^{3/2}/\pi$. Along the local vertical steepest-descent direction $t=t_0+iy$, the exponent is $S(t_0)-S^{\prime \prime }(t_0)y^2/2$ to quadratic order. The factor $(t/2\pi {)}^{12}$ varies negligibly over its Gaussian width. HenceThis proves the large-level degeneracy of an open bosonic string, including the otherwise unspecified leading constant. The factor $p^{-6}$ comes from the eta prefactor and $p^{-3/4}$ from the saddle width. As allowed, no bound on the remote part of the contour is needed.

At high mass, $p\simeq {\alpha }^{\prime }{M}^2$, so the density grows exponentially as $e^{{\beta }_{H}M}$ within units $k_B=\hslash =c=1$. A canonical sum with $e^{-\beta M}$ cannot converge for $\beta <{\beta }_H$, regardless of polynomial prefactors. This suggests a Hagedorn threshold and a breakdown or transition of the ordinary dilute-string thermal description. The power prefactor can make the partition sum itself finite at the threshold while sufficiently high energy moments diverge; the asymptotic count alone does not prove an absolute upper temperature. These are properties of the formal high-level spectrum, separate from the tachyon instability of the bosonic vacuum.