Solutions to the following equation including a Mobius function Let $\mu$ be the Mobius function. Now, how would one solve the following equation with respect to the variables $a\in \mathbb{N}$ and $b\in \mathbb{N}$?
$\sum_{d=1}^{\infty}\mu(d)\lfloor \frac{a}{d}\rfloor\lfloor \frac{b}{d}\rfloor = a\cdot b$?
Alternatively, how could we find the local maximums and minimums of the function $\sum_{d=1}^{\infty}\mu(d)\lfloor \frac{a}{d}\rfloor\lfloor \frac{b}{d}\rfloor$ with respect to $a$ or $b$ when we actually know the other variable? Can we do this by differentiation?  Is there any way of doing this?

All the questions you asked can be answered with the exact formula
\begin{align*}
\sum_{d=1}^\infty \mu(d) \lfloor \tfrac ad \rfloor \lfloor \tfrac bd \rfloor
&= \sum_{d=1}^\infty \mu(d) \sum_{\substack{m\le a \\ d\mid m}} 1 \sum_{\substack{n\le b \\ d\mid n}} 1 \\
&= \sum_{m\le a} \sum_{n\le b} \sum_{\substack{d\ge 1 \\ d\mid m \\ d\mid n}} \mu(d) \\
&= \sum_{m\le a} \sum_{n\le b} \sum_{\substack{d\ge 1 \\ d\mid\gcd(m,n)}} \mu(d) = \sum_{m\le a} \sum_{\substack{n\le b \\ \gcd(m,n)=1}} 1.
\end{align*}
(Definitely avoid differentiation with functions that aren't even continuous.)
