A new zero-density estimate for $ζ(s)$ and the error term in the Prime Number Theorem

We will provide a new type of zero-density estimate for $ζ(s)$ when $σ$ is sufficiently close to $1$. In particular, we will show that $N(σ,T)$ can be bounded by an absolute constant when $σ$ is sufficiently close to the left edge of the Korobov-Vinogradov zero-free region. As a consequence, we provide the optimal error term in the prime number theorem of the form $$ ψ(x)-x \ll x\exp \left\{-(1-\varepsilon) ω(x)\right\},\qquad ω(x):=\min _{t \geq 1}\{ν(t) \log x+\log t\}, $$ where $ν(t)=A_0(\log t)^{-2/3}(\log\log t)^{-1/3}$ is a decreasing function such that $ζ(σ+it)\neq 0$ for $σ\ge 1-ν(t)$. Precisely, we will show that we can take $\varepsilon=0$.