Guth–Maynard: $N(\sigma,T)\ll T^{\frac{30}{13}(1-\sigma)+\varepsilon}$

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 $\sigma\in[\tfrac12,1]$ and every $\varepsilon>0$,

This is the zero-density estimate of Guth and Maynard (2024), the closest known approach to the density hypothesis exponent $2(1-\sigma)$; it corresponds to the asymptotics for primes in short intervals of length $x^{17/30}$.

Formalization Note The source writes $N(\sigma,T)\le N^{\frac{30}{13}(1-\sigma)+\varepsilon}$; the base $N$ is read as $T$ and $\le$ as $\ll$. Counts are elements of $\mathbb N\cup\{\infty\}$ (cast to $[0,\infty]$), so the bound also asserts that the count is finite.