On $|{\rm Li}(x)-\pi(x)|$ and primes in short intervals, primes in short   intervals $x^{1/2}$ (II), and distribution of nontrivial zeros of the Riemann   zeta function
  Part One: First, we prove that given each natural number $x\geq10^{3}$ we have $$|{\rm Li}(x)-\pi(x)|\leq c\sqrt{x}\log x\texttt{ or } \pi(x)={\rm Li}(x)+O(\sqrt{x}\log x)$$ where $c$ is a constant greater than $1$ and less than $e$. Hence the Riemann Hypothesis is true according to the theorem proved by H. Koch in 1901. 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 $\beta$, let $\Phi(x):=\beta x^{1/2}$ for $x\geq x_{\beta}$ that satisfies $e(\log x_{\beta})^{3}/x_{\beta}^{0.0327283}=\beta$. Then there are $$\frac{\pi(x+\Phi(x))-\pi(x)}{\Phi(x)/\log x}=1+O(\frac{1}{\log x})$$ and $$\lim_{x \to \infty}\frac{\pi(x+\Phi(x))-\pi(x)}{\Phi(x)/\log x}=1.$$ Part Two: Based on a theorem for the numbers of primes in short intervals $x^{1/2}$, some conjectures of distribution of primes in short intervals, such as Legendre's conjecture, Oppermann's conjecture, Hanssner's conjecture, Brocard's conjecture, Andrica's conjecture, Sierpinski's conjecture and Sierpinski's conjecture of triangular numbers are proved and the Mills' constant can be determined. 
Shan-Guang Tan
math.GM