Huxley's zero-density estimate $N(\sigma,T)\ll T^{\frac{3}{3\sigma-1}(1-\sigma)+\varepsilon}$ for $\sigma\ge\tfrac34$

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. For every $\sigma\in[\tfrac34,1]$ and every $\varepsilon>0$,

This is Huxley's zero-density estimate (1971/72), unconditional; it improves on Ingham's exponent exactly when $\sigma>\tfrac34$.

Formalization Note The source does not state the range of $\sigma$; it is taken to be $\tfrac34\le\sigma\le1$, the range of Huxley's theorem (the two exponents coincide at $\sigma_{I,H}=\tfrac34$, as the source notes). Counts are elements of $\mathbb N\cup\{\infty\}$ (cast to $[0,\infty]$), so the bound also asserts that the count is finite.