= Solution
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 $M^2=(p-1)/\alpha'$. 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 $\sum_{r\ge0}w^{nr}=(1-w^n)^{-1}$. Therefore the <open-string oscillator degeneracy> generating function is
$$
F(w)=\sum_{p\ge0}d_pw^p=\prod_{n\ge1}(1-w^n)^{-24},
\qquad
\boxed{d_p=\frac1{2\pi i}\oint F(w)w^{-p-1}\,dw.}
$$
This 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 obtain
$$
\log F(e^{-t})=24\sum_{m\ge1}\frac1{m(e^{mt}-1)}.
$$
Since $t/(e^{mt}-1)\le1/m$, dominated convergence gives $t\log F(e^{-t})\to24\sum_{m\ge1}m^{-2}=4\pi^2$. Thus the leading singular term is $4\pi^2/t$.
The subleading term printed in the PDF also needs correction: the product implies $+12\log\epsilon$, rather than $-6\log\epsilon$. 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 https://dlmf.nist.gov/23.18[NIST's eta transformation formulas], and the product at large imaginary argument give
$$
\boxed{F(e^{-t})=\left(\frac{t}{2\pi}\right)^{12}
\exp\left(\frac{4\pi^2}{t}-t\right)
\left[1+O(e^{-4\pi^2/t})\right].}
$$
Since $t=-\log(1-\epsilon)=\epsilon+\epsilon^2/2+O(\epsilon^3)$, this yields
$$
\log F(1-\epsilon)=\frac{4\pi^2}{\epsilon}+12\log\epsilon
-12\log(2\pi)-2\pi^2+O(\epsilon).
$$
The 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 is
$$
d_p\simeq\frac1{2\pi i}\int\left(\frac{t}{2\pi}\right)^{12}e^{S(t)}\,dt,
\qquad S(t)=(p-1)t+\frac{4\pi^2}{t}.
$$
Set $q=p-1$. The positive saddle is $t_0=2\pi/\sqrt q$, with $S(t_0)=4\pi\sqrt q$ and $S''(t_0)=q^{3/2}/\pi$. Along the local vertical steepest-descent direction $t=t_0+iy$, the exponent is $S(t_0)-S''(t_0)y^2/2$ to quadratic order. The factor $(t/2\pi)^{12}$ varies negligibly over its Gaussian width. Hence
$$
d_p\sim\frac{(t_0/2\pi)^{12}}{\sqrt{2\pi S''(t_0)}}e^{S(t_0)}
=\frac1{\sqrt2}q^{-27/4}e^{4\pi\sqrt q}
\sim\boxed{\frac1{\sqrt2}p^{-27/4}e^{4\pi\sqrt p}.}
$$
This 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, $p\simeq\alpha'M^2$, so the density grows exponentially as $e^{\beta_HM}$ with
$$
\boxed{\beta_H=4\pi\sqrt{\alpha'},\qquad T_H=\frac1{4\pi\sqrt{\alpha'}}}
$$
in units $k_B=\hbar=c=1$. 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.
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