To begin, let us set

$$A_Q(n):=\sum_{d|n \\ d<Q}\mu(d)$$

If we fix $Q$ and let $n$ vary, we get a very surprising amount of cancellation. For instance, the trivial bound

\begin{align*} \mathbb{E}_{n\in\mathbb{N}}\left[|A_Q(n)|\right]&\leq\mathbb{E}_{n\in\mathbb{N}}\left[\sum_{\substack{d|n \\ d<Q}}1\right]\\ &=\sum_{d=1}^{Q-1}\frac{1}{d}\sim\log(Q) \end{align*}

can be reduced to the (astonishing) $\mathbb{E}_{n\in\mathbb{N}}[|A_Q(n)|]=O(1)$, and even more strongly $\mathbb{E}_{n\in\mathbb{N}}[|A_Q(n)|^2]=O(1)$. The question now turns the exact nature of the distribution of $A_Q(n)$ over $\mathbb{N}$ as $Q$ varies.

On the "maximum" side of things, the only trivial bound is using Sperner's Lemma which yields the inequality

$$\sup_{n\in\mathbb{N}}|A_Q(n)|\leq {\pi(Q)\choose \pi(Q)/2}\leq \frac{2^{\pi(Q)}}{\sqrt{\pi(Q)}}$$

Stronger results require finer knowledge of the distribution of primes, and more specifically getting a large value $A_Q(n)$ means that the prime factors of $n$ are tightly packed and so large values of $A_Q(n)$ are morally equivalent to repeated small prime gaps (i.e a reltively small interval $M$ in which many primes appear). While proving any results seem difficult, numerical evidence suggests the following miraculous asymptotic relationship

$$\sup_{n\in\mathbb{N}}|A_Q(n)|\sim_{Q\to\infty}\pi(Q)$$

There are other seemingly miraculous properties of $A_Q(n)$, like the fact that the probabilities $\Pr_{n\in\mathbb{N}}[A_Q(n)=j]$ seem to converge as $Q\to\infty$ for any $j$, but that is a different can of worms entirely.

If somebody could give any insights about how someone would even try to go about proving $\sup_{n\in\mathbb{N}}|A_Q(n)|\sim_{Q\to\infty}\pi(Q)$ I would be extremely grateful, since currently all of my approaches seem to only result in weak upper bounds.

the only trivial bound is using Sperner's Lemma...I would say "the most trivial bound is $Q$: at most $Q$ terms each of which is at most $1$". The bound $\pi(Q)$ is, of course not so immediate... $\endgroup$ – fedja Jan 10 at 2:43