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I have come across the statement that $$ \sum_{n\leq x}\sum_{d\mid(n,P_z)}\mu(d) = \sum_{d\mid P_z}\mu(d) \left[\frac{x}{d}\right], $$ where $P_z=\prod_{p\leq z}p$ where $p$ is prime, $\mu(d)$ is the Mobius function, and $[x]$ denotes the greatest integer less than or equal to $x$. I would appreicate some help in seeing the above. (Both of the above are equal to $\pi(x,z)$, the number of (positive) integers less than or equal to $x$ which are prime to all primes less than or equal to $z$.)

I am aware of the property that $\sum_{d\mid n}\mu(d)=0$ unless $n=1$ in which case, the sum is equal to $1$. I can see how for a given $d$, there are $[x/d]$ integers less than or equal to $x$ with this divisor.

Thank you!

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For every $d$, count how often $\mu(d)$ occurs in the sum on the left.

If it occurs at all, there is an $n\leqslant x$ with $d\mid (n,P_z)$. Since $(n,P_z) \leqslant n \leqslant x$, it follows that $d\leqslant x$. Also, since $d$ divides $(n,P_z)$, it divides $P_z$.

Now, if $d \leqslant x$ and $d\mid P_z$, it occurs once in the inner sum for every $n \leqslant x$ such that $d \mid n$. There are exactly $\left[\frac{x}{d}\right]$ such $n$. For the divisors of $P_z$ that are greater than $x$, we have $\left[\frac{x}{d}\right] = 0$, and above we saw that $\mu(d)$ does not occur in the sum on the left for these $d$, so $\mu(d)$ occurs $\left[\frac{x}{d}\right]$ times in the sum on the left for all divisors of $P_z$, whether $\leqslant x$ or $> x$, i.e.

$$\sum_{n\leqslant x} \sum_{d\mid (n,P_z)} \mu(d) = \sum_{d\mid P_z} \mu(d)\left[\frac{x}{d}\right].$$

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