Best available bounds for $\pi(Y)-\pi(Y-X)$?

Question

I don't know much (anything) about sieves, but as I read the section on the Selberg upper bound sieve from Greaves's Sieves in Number Theory, there is a theorem 4 which says that If $Y\ge X \ge 2$, then

\begin{equation} \pi(Y)-\pi(Y-X) \leq \frac{2X}{\log X} + O\left(\frac{X}{\log^2 X}\right). \end{equation}

1. I would be interested in knowing what the best available bound is.
2. Also, I notice that this bound holds irrespective of the value of $Y$. Are there ways to improve it for say $Y = 2X$ or $X= \sqrt{Y}$?

Thank you in advance!

Here, $\pi(X) = \sum_{p\leq x} 1$.

Answer 1

If $Y\geq X\geq 2$, then as Daniel Johnston wrote, Montgomery and Vaughan proved that

$$\pi(Y)-\pi(Y-X)\leq \frac{2X}{\log X}.$$

Whether this constitutes a "best bound" requires a definition of what you consider to be "best". Expounding on Noam Elkies' comment, it follows from work of Heath-Brown (building on Hoheisel, Tchudakov, Heath-Brown and Iwaniec, Huxley, and many others) that if $Y^{7/12}\leq X\leq Y$, then

$$\pi(Y)-\pi(Y-X)\sim\frac{X}{\log Y}.$$

Therefore, for all $\epsilon>0$ there exists $Y_0(\epsilon)>0$ such that if $Y\geq Y_0(\epsilon)$ and $Y^{7/12}\leq X\leq Y$, then

$$\Big|\pi(Y)-\pi(Y-X)-\frac{X}{\log Y}\Big|\leq \epsilon \frac{X}{\log Y}.$$

(This can be made more precise using known error terms in the prime number theorem for short intervals.) The notion of "best" depends on context.