Gaps between critical zeros of the Riemann zeta-function

Let $\rho = \beta + i\gamma$ denote a nontrivial zero of the Riemann zeta-function, $\zeta(s)$, and consider the sequence of ordinates of these zeros in the upper half-plane: $0<\gamma_1 \le \gamma_2 \le \cdots \le \gamma_n \le \cdots$. Since the number of zeros up to height $T$ is about $T/(2\pi \log T)$, the average spacing between the consecutive zeros $\gamma_{n+1}$ and $\gamma_n$ is $2\pi / \log \gamma_n$. In this lecture, we will study small and large (normalized) gaps between zeros of $\zeta(s)$, primarily focusing on the quantities
\[
\mu:= \liminf_{n\to\infty}\frac{\gamma_{n+1} - \gamma_n}{2\pi / \log \gamma_n}, \qquad \lambda:= \limsup_{n\to\infty}\frac{\gamma_{n+1} - \gamma_n}{2\pi / \log \gamma_n}.
\] 
By definition $\mu \le 1 \le \lambda$, but it is expected that $\mu = 0$ and $\lambda = \infty$, i.e. it is expected that there are arbitrarily small and large (normalized) gaps between consecutive zeros of $\zeta(s)$. We will first explore the origin of these conjectures via Montgomery's work on the pair correlation of zeros of $\zeta(s)$, and then we will turn our attention to the current methods used to give the strongest bounds on $\mu$ and $\lambda$. We will also discuss the limitations of these methods and the problem's connections to the class number problem for imaginary quadratic fields and moments of the Riemann zeta-function.