标题： 一个新的零密度估计用于$ζ(s)$以及素数定理中的误差项 显示英文标题

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

摘要： 我们将为$\zeta(s)$提供一种新的零密度估计，当$\sigma$足够接近$1$时。特别地，我们将证明当$\sigma$足够接近 Korobov-Vinogradov 零点自由区域的左侧边缘时，$N(\sigma,T)$可以被一个绝对常数所界定。 因此，我们提供了素数定理中最优的误差项，形式为$$ \psi(x)-x \ll x\exp \left\{-(1-\varepsilon) \omega(x)\right\},\qquad \omega(x):=\min _{t \geq 1}\{\nu(t) \log x+\log t\}, $$，其中$\nu(t)=A_0(\log t)^{-2/3}(\log\log t)^{-1/3}$是一个递减函数，使得$\zeta(\sigma+it)\neq 0$对于$\sigma\ge 1-\nu(t)$成立。 确切地说，我们将证明我们可以取$\varepsilon=0$。 显示英文摘要

摘要： 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 $$ \psi(x)-x \ll x\exp \left\{-(1-\varepsilon) \omega(x)\right\},\qquad \omega(x):=\min _{t \geq 1}\{\nu(t) \log x+\log t\}, $$ where $\nu(t)=A_0(\log t)^{-2/3}(\log\log t)^{-1/3}$ is a decreasing function such that $\zeta(\sigma+it)\neq 0$ for $\sigma\ge 1-\nu(t)$. Precisely, we will show that we can take $\varepsilon=0$.