Assume an arbitrary $x$ and let $z$ be smaller than $y$, where $y$ is the length of the interval $[x,x+y]$. What I would like to know is:

Let $W(z)=\prod_{p\leq z}\left(1-\frac{1}{p}\right)$. For what values of $z=z(y)$ is $$ y W(z) \sim \textrm{e}^{-\gamma} \frac{y}{\log z} $$ an accurate estimate of the number of integers coprime to all primes $p\leq z$ in the interval $[x,x+y]$?

A trivial answer—since the set of coprimes to $p\leq z$ is periodic with period $P(z)=\prod_{p\leq z} p$—is to choose $z$ so that $y/P(z) p \rightarrow \infty$. From numerical experiments, however, it seems that the far more moderate constraint $z/y\rightarrow 0$ is sufficient. So when can we indeed assume that Merten's product theorem correctly estimates the number of coprimes in an interval?

**EDIT 1:** To be specific, the observation I make numerically is that if we look at the interval $[x,x+y]$ in regions where $x$ is much larger than $y$, but still much smaller than $P(z)$, Merten's product theorem seems to accurately describe the density of coprimes to $p<z$ in $[x,x+y]$, even when $(\log y) /(\log z)=u$ is constant. Take for example $y=10^5$, $z=y^{1/2}$, and $x=n\cdot10^8$, where $n=1,\dots,512$. Then the mean density of coprimes to $p<z$ in $[x,x+y]$ ($\pm$ standard deviation) takes the value
$$
(0.9998(5) \pm 0.0026(1))\cdot W(z).
$$
Larger values of $u$ produce even more precise values, so e.g. $z=y^{1/3}$ for the same $x$ and $y$ results in mean density
$$
(0.99999(9)\pm 0.00050(6)) \cdot W(z),
$$
and likewise for $z=y^{1/4}$,
$$
(0.99999(7)\pm 0.00011(6)) \cdot W(z).
$$

From these numerical experiments the conjectured behavior is that if $z=y^{1/u}$, where $u>1$ is a constant, and we let $y\rightarrow \infty$, the density of coprimes to $p<z$ in $[x,x+y]$ is accurately approximated by $\textrm{e}^{-\gamma}/\log z$, given that $x$ is much larger than $y$ and much smaller than $P(z)$.

I'm interested in learning whether anyone has good heuristic arguments as to why and when these observations hold, and if possible, some more rigorous footing.

(Note that in order to emphasize the accuracy of the measurements, I have expressed the density values in terms of W(z) rather than $\textrm{e}^{-\gamma}/\log z$, as this would include a hidden error term.)

**EDIT 2:** While the numerical standard deviations above appear small, we could perhaps expect a larger standard deviation if we consider the number of coprimes to $P(z)$ in a random interval of length $y$. This poses a couple of questions (where we no longer consider $z$ to relate to $y$):

Q1: Is there a known upper bound on the variance of the number of coprimes to $P(z)$ for a randomly selected interval of length $y$?

Q2: Is the variance of the binomial distribution $B(y,W(z))$, given by $y W(z)(1-W(z))$, an upper bound of the variance of the number of coprimes to $P(z)$ in a random interval of length $y$?

The last question is motivated by the fact that in the extreme case when $y=1$, the probability of finding a coprime reduces to that of a Bernoulli trial with variance $W(z)(1-W(z))$. But for $y>1$, the probabilities of finding coprimes along $y$ are no longer independent. Furthermore, the variance is necessarily periodic with period $P(z)$. It therefore seems plausible that the binomial distribution $B(y,W(z))$ has a variance that is always greater than that of counting coprimes to $P(z)$ in a random interval of any length $y$.