素数定理的显式界
Explicit bounds for the prime number theorem
- University of Lethbridge（莱斯布里奇大学）
- University of Turku（图尔库大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过多种零区域和零密度估计，为$L$-函数零点和的显式界提供新方法，并应用于黎曼zeta函数，得到素数定理误差项的新最佳估计。
AI中文摘要：
我们研究在多种零区域和零密度估计下，寻找$L$-函数零点在$1$-线附近和的紧显式界的问题。作为展示这些技术的应用，我们将其应用于黎曼zeta函数和素数定理的情形，从而在素数定理的误差项上获得新的最佳估计。我们证明，对于$x\geq21$，有$|\psi(x)-x| < 0.239 x \exp\left(-0.1982767\left(1+\frac{\log\log\log(x)}{15\log\log(x)}\right) \frac{\log(x)^{3/5}}{\log\log(x)^{1/5}}\right)$。
英文摘要：
We study the problem of finding tight explicit bounds on sums over zeros of $L$-functions near the $1$-line with a variety of zero-free regions and zero-density estimates. As an application to demonstrate the techniques, we apply them to the case of the Riemann zeta function and the prime number theorem to obtain new best estimates on the error terms in the prime number theorem. We show that for $x\geq21$ we have $|ψ(x)-x| < 0.239 x \exp\left(-0.1982767\left(1+\frac{\log\log\log(x)}{15\log\log(x)}\right) \frac{\log(x)^{3/5}}{\log\log(x)^{1/5}}\right).$
