I can't understand well the proof of theorem 13.3
There exist a constant $C>0$ s.t. if RH is true, then for every $x\ge 2$ the interval $(x,x+Cx^{1/2}\log x)$ contains at least $x^{1/2}$ prime numbers.
I understand the formula
$$\sum_{n}\Lambda(n)\omega(n)=\frac{1}{h}(\psi_1 (x+h)-2\psi_1 (x)+\psi_1 (x-h))=h-\frac{1}{h}\sum_{\rho}\frac{(x+h)^{\rho+1}-2x^{\rho+1}+(x-h)^{\rho+1}}{(\rho)(\rho+1)}+O(\frac{1}{hx})$$
but I can't understand writer's explanation under this formula.
Can you give me a more detailed proof of this theorem please?