= {\frac}{j}{p}$$

The Thouless formula holds: $${\label}{2.16} 
\gamma(z) =\int \log{\lvertz-x\rvert}\, dk(x) + \log C_B$$ where $\gamma$ is given by and $C_B$ is the (logarithmic) capacity of $B$.

The (logarithmic) capacity of the bands is given by $${\label}{2.17} 
C_B = \biggl(\, \prod_{j=1}^p a_j\biggr)^{-1}$$

$${\label}{2.18x} 
\Gamma_+(z) =C_B \exp \biggl( p \int \log (z-x)\, dk(x) \biggr)$$

That completes the review of periodic OPRL. We now turn to the study of the zeros. We begin by describing exactly (not just asymptotically!) the zeros of $P_{mp-1}$:

[T2.1]{} The zeros of $P_{mp-1}(x)$ are exactly

The $p-1$ Dirichlet data points $\{x_j\}_{j=1}^{p-1}$.

The $(m-1)p$ points $\{x_{k,q}^{(m)}
\}_{\substack{k=1, \dots, p \\ q=1, \dots, m-1}}$ where $${\label}{2.18} 
k(x_{k,q}^{(m)}) = {\frac}{k-1}{p} + {\frac}{q}{mp}$$

[*Remarks.*]{} 1. The points of can be described as follows. Break each band $[\alpha_j,\beta_j]$ into $m$ pieces of equal size in equilibrium measure. The $x_{k,q}^{(m)}$ are the interior break points.

2\. If a gap is closed, we include its position in the “Dirichlet points" of (i).

3\. Generically, there are not zeros at the band edges, that is, has $q=1, 
\dots, m-1$ but not $q=0$ or $q=m$. But it can happen that one or more of the Dirichlet data points is at an $\alpha_{j+1}$ or a $\beta_j$.

4\. 