Estimating the number of integers in a sequence of consecutive integers that are relatively prime to a given primorial Let $x,y$ be positive integers and $p$ a prime.
Is there a standard way to estimate the number of integers $z$ where  $x \le z < x+y$ and $\gcd(z,p\#)=1$
For example, for $x=1000, y=30, p=7$, there are $7$ integers in the range $1000 \le z < 1030$ that are relatively prime to $210$ which are $\left\{ 1003, 1007, 1009, 1013, 1019, 1021, 1027\right\}$
 A: I don't know of any "standard" way to estimate the number of relatively prime integers. The best general way I know of is to assume the integers relatively prime to $p\#$ are roughly equidistributed. Since there are $\varphi(p\#)$ (i.e., using Euler's totient function) of them in $[1, p\#]$, this gives an estimate of
$$r(y, p) = \frac{y(\varphi(p\#))}{p\#} \tag{1}\label{eq1A}$$
With your example, this becomes
$$\begin{equation}\begin{aligned}
r(30, 7) & = \frac{30(\varphi(7\#))}{7\#} \\
& = \frac{30(6(4)(2))}{210} \\
& = \frac{48}{7}
\end{aligned}\end{equation}\tag{2}\label{eq2A}$$
The estimate is just $\frac{1}{7}$ less than the correct value of $7$. With just varying $x$, the estimate in \eqref{eq2A} will be quite good, being off by no more than about $1$ or so (note, though, I haven't checked exactly what the maximum error is). For example, with $x = 700$, there are $6$ relatively prime integers of $\{701, 703, 709, 713, 719, 727\}$, so the estimate is then $\frac{6}{7}$ too large.
For any given $y$ and $p$, the actual value depends on just $x \bmod{p\#}$. For example, there are $7$ integers in $[0, 29]$ which are relatively prime to $210$, i.e., $\{1, 11, 13, 17, 19, 23, 29\}$, so for $x$ being any multiple of $210$, there'll likewise always be $7$ relatively prime integers in $[x, x + 29]$.
However, the error can be somewhat larger, such as in some cases where $y$ is fairly small compared to $p\#$ and, especially, where $y$ mostly just covers the smaller integers modulo $p\#$ (e.g., when $x$ is just over a multiple of $p\#$) since most of the smaller integers have a prime factor $\le p$. Using $p_n$ being the $n$'th prime, a fairly extreme example is there's no integer in $[2, p_{n+1} - 1]$ which is relatively prime to $p_n\#$. For example, keeping $p = 7$, but with $x = 212$ and $y = 9$, there are no relatively prime numbers in $[212, 220]$, but the estimate from \eqref{eq1A} is
$$\begin{equation}\begin{aligned}
r(9, 7) & = \frac{9(\varphi(7\#))}{7\#} \\
& = \frac{9(6(4)(2))}{210} \\
& = \frac{72}{35}
\end{aligned}\end{equation}\tag{3}\label{eq3A}$$
i.e., just over $2$. The maximum absolute error grows fairly slowly compared to $p\#$, but nonetheless it's unbounded. However, especially for cases where $y$ is considerably larger than $p\#$, the maximum error relative to $y$ will always be small, with this relative error approaching $0$ as $y$ grows.
Depending on your needs, \eqref{eq1A} may be sufficiently accurate. If you require something more precise, you can enhance \eqref{eq1A} to include $x$ and handle various cases to better deal with particular situations, like the result in \eqref{eq3A}. Since I don't know if you require this, and if you do to what extent and what would then be the best way to deal with it, I'll leave it to you to handle this yourself as need be.
