TITLE: Prove or disprove that $\sup_{n\in\mathbb{N}}\left|\sum_{\substack{d|n \\d<Q}}\mu(d)\right|\sim\pi(Q)$
QUESTION [6 upvotes]: 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.

REPLY [12 votes]: 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 (see Theorem 5 there) and the lower bound is due to Hall and Tenenbaum (see the references in Montgomery and Vaughan).