Let $\Phi(x,p)$ = the number of integers $i$ where $0 < i \le x$ and $\gcd(x,p)=1$.

Let $p\#$ be the primorial for $p$.

Using the inclusion-exclusion principle:

$$\Phi(x,p\#) = \sum_{d|p\#}\mu(d)\left\lfloor\frac{x}{d}\right\rfloor$$

If I apply the Heilbronn-Rohrbach Inequality, does it follow that:

$$\left\lfloor\left(\prod_{q \le p \text{ and q prime}}\frac{q-1}{q}\right)x\right\rfloor \le \sum_{d|p\#}\mu(d)\left\lfloor\frac{x}{d}\right\rfloor \le \left\lceil\left(\prod_{q \le p \text{ and q prime}}\frac{q-1}{q}\right)x\right\rceil$$

I am asking this question because the product is much easier to work with than the sum.

At the same time, my suspicion is that this inequality may be too good to be true. In which case, I am very interested in understanding the circumstances where it is not correct.