Under Lindelöf, the density hypothesis $N(\sigma,T)\ll T^{2(1-\sigma)+\varepsilon}$ holds

Let $N(\sigma,T)$ be the number of zeros $\rho$ of $\zeta$ with $\operatorname{Re}\rho\ge\sigma$ and $|\operatorname{Im}\rho|\le T$, counted with multiplicity. Assume the Lindelöf hypothesis. Then for every $\sigma\in[\tfrac12,1]$ and every $\varepsilon>0$,

This is the density hypothesis, which the source records as following from the Lindelöf hypothesis.

Formalization Note RiemannVonMangoldt.LindelofHypothesis is the published definition: for every $\varepsilon>0$, $\zeta(\tfrac12+it)=O(t^{\varepsilon})$ as $t\to+\infty$. The source writes $N(\sigma,T)\le N^{2(1-\sigma)+\varepsilon}$; the base $N$ is read as $T$, the standard form of the density hypothesis, and $\le$ as $\ll$ (an implied constant, $T$ large). The range $\tfrac12\le\sigma\le1$ is the standard range of the density hypothesis. Counts are elements of $\mathbb N\cup\{\infty\}$ (cast to $[0,\infty]$), so the bound also asserts that the count is finite.