by,

$$\begin{aligned}
\label{eq:Fnp}
\widetilde{F}_2^{np}(x) = -2\sum_{n,m=1}^{\infty}f_{nm}(x).\end{aligned}$$

For the $SU(N)$ partition function we divide by the $U(1)$ part (assuming only commutator interactions so the $U(1)$ part is decoupled from the $SU(N)$ part of the theory),

$$\begin{aligned}
\label{eq:SUnZ}
\cal{Z}_{SU(N)}=\frac{\cal{Z}^{U(N)}}{\cal{Z}^{U(1)}}=e^{-\lambda\beta \tilde{F}_2^{np}}\prod_{n=1}^{\infty}\frac{e^{-z_n(x)/n+\lambda\beta f_n(x)}}{1-z_n(x)+\lambda n\beta f_n(x)}.\end{aligned}$$

The goal of our computation is therefore to compute $f_n(x)$ and $f_{nm}(x)$.

The Hagedorn temperature is defined as the temperature for which the large $N$ partition function diverges. This happens when one of the denominators in (\[eq:UnZ\]) or (\[eq:SUnZ\]) vanishes. Since the 1-particle partition sums are monotonic functions with $z(0)=0$ and $z(1)=\infty$ this always occurs first for the $n=1$ denominator in the multiplication of (\[eq:UnZ\]) or (\[eq:SUnZ\]), so the Hagedorn temperature is given by $z_1(x_H)=1$ . The order $\lambda$ correction to the Hagedorn temperature is therefore,

$$\begin{aligned}
\label{eq:HagedornCorrection}
\delta x_H = \beta_H\lambda\frac{f_1(x_H)}{z'_1(x_H)},\end{aligned}$$

where $x_H$ is the Hagedorn temperature of the free theory.

Other quantities that can be extracted from the partition function are the sums of the order $\lambda$ corrections to the spectrum of **YM** on $S^3\times \mathds{R}$. These can be obtained by expanding the partition function in small $x$ (small temperatures)[^6],

$$\begin{aligned}
\label{eq:SmallxExp}
\cal{Z}_{SU(N)} = \sum_{\textrm{all states}}x^{E_i+\lambda\delta E_{i}}=\sum_{n=4}^{\infty}x^{\frac{n}{2}}\left(d\left(\frac{n}{2}\right)+\lambda\ln(x)\sum_i\delta E_{n,i}\right),\end{aligned}$$

where in the second equality the sum is over 