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.



Prove or disprove that $\sup_{n\in\mathbb{N}}\left|\sum_{\substack{d|n \\d<Q}}\mu(d)\right|\sim\pi(Q)$
This is not true. In fact
$$ 
x(\log x)^{-1+1/\pi} \gg \sup\_n \Big| \sum\_{\substack{ d|n \\ d\le x}} \mu(d) \Big| \gg x (\log x)^{-1+1/\pi}. 
$$
The upper bound is due to [Montgomery and Vaughan](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/43188/10998_2004_Article_400315.pdf?sequence=1&isAllowed=y) (see Theorem 5 there) and the lower bound is due to Hall and Tenenbaum (see the references in Montgomery and Vaughan).


