To add to GH from MO's answer, Chapter 10 of Iwaniec and Kowalski's "Analytic Number Theory" is another good reference. A few of the classical papers are

- Ingham's "On the Estimation of $N(\sigma,T)$,"
- Montgomery's "Mean and Large Values of Dirichlet Polynomials" and "Zeros of $L$-functions," and
- Huxley's "On the Difference Between Consecutive Primes."

Define
$$
N(\sigma,T) = |\left\{\rho=\beta+i\gamma: \zeta(\rho)=0,\ \beta\geq \sigma,\ 0\leq \gamma \leq T\right\}|,
$$
so $N(\sigma,T)$ counts the zeros of $\zeta$ with real part at least $\sigma$ and imaginary part at most $T$. In general, zero density estimates for the Riemann zeta function take the form
$$
\tag{1}
N(\sigma,T) \ll T^{\lambda(\sigma)(1-\sigma)}(\log T)^B
$$
for some relatively unimportant positive constant $B$ and some function $\lambda(\sigma)$. Note that the bound shrinks as $\sigma \to 1$. Ingham shows that (1) holds with
$$
\lambda(\sigma) = \frac{3}{2-\sigma}.
$$
Montgomery proves an analogous result for all Dirichlet $L$-functions and also proves that (1) holds with
$$
\lambda(\sigma) = \frac{5}{2},
$$
which Huxley improves to
$$
\lambda(\sigma) = \frac{12}{5}.
$$
Both Montgomery and Huxley prove results slightly stronger than what I've stated. The *density conjecture* states that one can take $\lambda(\sigma)=2$, but this is only known in limited ranges of $\sigma$. For instance, $\sigma \geq \frac{5}{6}$ follows from Huxley's work. The full range would follow from the Riemann Hypothesis.

Zero density estimates are connected to the distribution of primes in short intervals. Ingham, in a paper with the same name as Huxley's, shows that any estimate of the form (1) with $\lambda$ constant implies that any interval $[x,x+x^\theta]$ with $\theta > 1-\lambda^{-1}$ and $x$ sufficiently large contains a prime.

The general strategy for proving zero density estimates is to construct "zero detecting polynomials," which are functions that take unusually large values at the zeros of $\zeta$. One then uses "mean and large value estimates" to show that these functions cannot be large too often.