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

We will provide a new type of zero-density estimate for $\zeta (s)$ when $\sigma$ is sufficiently close to $1$. In particular, we will show that $N(\sigma ,T)$ can be bounded by an absolute constant when $\sigma$ 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 $$\psi (x)-x\ll x\exp \{-(1-\epsilon )\omega (x)\},\omega (x):=\underset{t\ge 1}{\min }\{\nu (t)\log x+\log t\},$$ where $\nu (t)=A_0(\log t{)}^{-2\mathrm{/}3}(\log \log t{)}^{-1\mathrm{/}3}$ is a decreasing function such that $\zeta (\sigma +it)\mathrm{\ne }0$ for $\sigma \ge 1-\nu (t)$. Precisely, we will show that we can take $\epsilon =0$.