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Part One: First, we prove that given each natural number $x\geq10^{3}$ wehave $$|{\rm Li}(x)-\pi(x)|\leq c\sqrt{x}\log x\texttt{ or } \pi(x)={\rmLi}(x)+O(\sqrt{x}\log x)$$ where $c$ is a constant greater than $1$ and lessthan $e$. Hence the Riemann Hypothesis is true according to the theorem provedby H. Koch in 1901. Second, with a much more accurate estimation of primenumbers, 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 $x\geqx_{\beta}$ that satisfies $e(\log x_{\beta})^{3}/x_{\beta}^{0.0327283}=\beta$.Then there are $$\frac{\pi(x+\Phi(x))-\pi(x)}{\Phi(x)/\log x}=1+O(\frac{1}{\logx})$$ and $$\lim_{x \to \infty}\frac{\pi(x+\Phi(x))-\pi(x)}{\Phi(x)/\logx}=1.$$ Part Two: Based on a theorem for the numbers of primes in shortintervals $x^{1/2}$, some conjectures of distribution of primes in shortintervals, such as Legendre's conjecture, Oppermann's conjecture, Hanssner'sconjecture, Brocard's conjecture, Andrica's conjecture, Sierpinski's conjectureand Sierpinski's conjecture of triangular numbers are proved and the Mills'constant can be determined.

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