\section{The Usual Counterterm}\label{sec:counterterm/spherical}

The spherical integrals $I_D$ are cut off by a radius of $N/2$.
In \eqref{spherical quantization} a variety of dimensionful parameters are separated from the integral itself.
For convenience, the integrals that appear in $S_D^\spherical$ for $x>0$ itself are given by
\begin{equation}
    \label{eq:spherical S improved regulator}
     \Omega_D \int_0^{N/2}  \mathcal P \frac{ n^{D-1} \mathrm{d}n}{n^2-x} = \begin{cases}
        4\pi\left( \frac{N}{2} - \sqrt{x} \tanh\inverse \frac{\sqrt{x}}{N/2}\right)
            &   (D=3)\\[0.25cm]
        \pi \log\left( \frac{(N/2)^2}{x} - 1\right)
            &   (D=2)\\[0.25cm]
        -\frac{2}{\sqrt{x}} \tanh\inverse \frac{\sqrt{x}}{N/2}                     &   (D=1)
    \end{cases}.
\end{equation}

The regulating behavior needed to cancel the divergence in the sum in $S_D^\spherical$ can be found by expanding these integrals around the large-$N$ behavior,
\begin{equation}
     \Omega_D\int_0^{N/2} \mathcal P \frac{ n^{D-1} \mathrm{d}n}{n^2-x} \goesto \begin{cases}
        4\pi \left(\frac{N}{2}\right) - \frac{4 \pi x}{N/2} + \order{\left(\frac{N}{2}\right)^{-2}}
            &   (D=3)\\[0.25cm]
        \pi \log \left(\frac{(N/2)^2}{x}\right) - \frac{\pi x}{(N/2)^2} + \order{\left(\frac{N}{2}\right)^{-4}}
            &   (D=2)\\[0.25cm]
        -\frac{2}{(N/2)} - \frac{2 x}{3 (N/2)^3} + \order{\left(\frac{N}{2}\right)^{-4}}
            &   (D=1)
    \end{cases}
\end{equation}
However, rather than including only the leading divergent behavior, one can use the exact integral values in \eqref{spherical S improved regulator}, accelerating the convergence to the large-$N$ limit.

In two dimensions, we can rewrite separate the large-$N$ behavior of the infinite-volume integral
\begin{align}
    \Omega_D\int_0^{N/2} \mathcal P \frac{ n^{D-1} \mathrm{d}n}{n^2-x}
    &=
    \pi \log \left(\frac{(N/2)^2}{x}\right) + \pi \log\left(1-\frac{x}{(N/2)^2}\right)
    &
    (D&=2)
\end{align}
but cannot separate out an $x$-independent counterterm.
