Let $N(\sigma,T)$ denote the number of zeros $\rho=\beta+\gamma i$ of the Riemann zeta function satisfying $\beta\ge \sigma$ and $0<\gamma\le T$, counted with multiplicity. Then the "Density Hypothesis" usually means one of the two following unproven conjectures.

**1."Weak" Density Hypothesis:**
Let $\varepsilon>0$ be fixed. Then
$$N(\sigma,T)=O(T^{2(1-\sigma)+\varepsilon})$$
uniformly in $1/2\le \sigma\le 1$ as $T\to +\infty$.

**2. "Strong" Density Hypothesis:** There exists a real number $\alpha\ge 1$, such that
$$N(\sigma,T)=O(T^{2(1-\sigma)}(\log T)^{\alpha})$$
uniformly in $1/2\le \sigma\le 1$ as $T\to +\infty$.

From the work of Hoheisel, it is known that if the (Strong) density hypothesis is true, then \begin{align} \psi(x+x^{\vartheta})-\psi(x)\sim&\;x^{\vartheta} &&(1)\\[1mm] \pi(x+x^{\vartheta})-\pi(x)\sim&\; \min(1,\vartheta^{-1})\frac{x^{\vartheta}}{\log x} &&(2)\\[1mm] p_{n+1}-p_{n}=&\;O(p_n^{\vartheta}) &&(3) \end{align} all hold for any fixed $\vartheta>1/2$.

**Question:** Are there any other (at least remotely significant) consequences that would follow if one of the above density hypotheses could be proven (without assuming the Lindelöf or Riemann Hypotheses)? $(1)-(3)$ surely cannot be an exhaustive list.

**Some background:** The density hypothesis arose from A.E. Ingham's paper *On the difference between consecutive primes* (1937), where he showed that, if $\zeta(\tfrac{1}{2}+ti)=O(t^{c})$ as $t\to +\infty$, for some fixed $c>0$, then
$$N(\sigma,T)=O(T^{2(1+2c)(1-\sigma)}(\log T)^{5})$$
uniformly in $1/2\le \sigma \le 1$ as $T\to +\infty$.

Thus, if the Lindelöf hypothesis is true, then we may take $c=\varepsilon$ and conclude that the weak density hypothesis is true. As explained in this post, the linear term $2(1-\sigma)$ is the best possible (or, at the very least, naturally occuring). As argued in the same post, the density hypothesis has been proven unconditionally for $\sigma \ge 25/32$ by Bourgain, although the conclusion over this range is somewhat stronger than the density hypothesis itself.

Supposing for the moment that we could in fact prove that the density hypothesis is true, then what would we gain from it?

In Ivić's book, it sais on p. 47 that "Estimates of $N(\sigma,T)$" play an important role in many applications of zeta-function theory, and for some applications concerning primes the reader is referred to chapter 12." Now $(1)-(3)$ are basically the results he is referring to. So where else are estimates of $N(\sigma,T)$ important to zeta-function theory, especially if the density hypothesis is true?