On $|{\rm Li}(x)-\pi(x)|$ and primes in short intervals

Abstract

Two topics of the number theory are discussed in this paper. First, we prove that given each natural number $x\geq10^{3}$, we have \[ |{\rm Li}(x)-\pi(x)|\leq c\sqrt{x}\log x\texttt{ and } \pi(x)={\rm Li}(x)+O(\sqrt{x}\log x) \] where $c$ is a constant greater than $1$ and less than $e$. Second, with a much more accurate estimation of prime numbers, the error range of which is less than $x^{1/2-0.0327283}$ for $x\geq10^{41}$, we prove a theorem of the number of primes in short intervals: Given a positive real number $\beta$ that determines a real number $x_{\beta}$ by $e(\log x_{\beta})^{3}/x_{\beta}^{0.0327283}=\beta$, let $\Phi(x):=\beta x^{1/2}$ for $x\geq x_{\beta}$ where $\Phi(x):=x^{1/2}$ when let $\beta=1$. Then there are \[ \frac{\pi(x+\Phi(x))-\pi(x)}{\Phi(x)/\log x}=1+O(\frac{1}{\log x}) \] and \[ \lim_{x \to \infty}\frac{\pi(x+\Phi(x))-\pi(x)}{\Phi(x)/\log x}=1. \]