  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.
