big $\mathcal O$ for number of prime in an interval?

Question

According to von Koch 1991, if the Riemann hypothesis is true, then the for the prime counting function

$$\pi(x)=Li(x)+\mathcal O(\sqrt x \log x)$$

I am trying to understand how to deal with the big $\mathcal O$ terms when estimating the number of primes within an interval $[y,z]$:

$$\Delta\pi=\pi(z)-\pi(y)=Li(z)-Li(y)+\mathcal O(\sqrt z \log z)-\mathcal O(\sqrt y \log y)$$

$$\Delta\pi=\int_y^z\frac{1}{\log x}dx+\underbrace{\mathcal O(\sqrt z \log z)-\mathcal O(\sqrt y \log y)}_?$$

How does the difference of the big $\mathcal O$ behave? Any references also welcome.

Many thanks in advance.

Answer 1

The tricky bit about the $O$ terms is that they do not represent numbers, but really sets, or if you prefer, families of functions. Further, they are only concerned with the modulus or absolute value, and hence $+$ and $-$ are simply the same.

In short: $O(\sqrt{z}\log z)-O(\sqrt{y}\log y)=O(\sqrt{z}\log z)$, that's all you can say about it.