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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.

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    $\begingroup$ $A_Q(\prod_{p\in (Q^{1/2},Q)} p) = 1-(\pi(Q-1)-\pi(Q^{1/2}))$ $\endgroup$ – reuns Jan 10 at 1:35
  • $\begingroup$ 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
  • $\begingroup$ @fedja are you sure about that? The $\pi(Q)$ comes from the fact that the sum $\sum_{\substack{d|n \\ d<Q}}\mu(d)$ is a sum over $2^{\pi(Q)}$ points, for quite immediate reasons... $\endgroup$ – Milo Moses Jan 10 at 6:07
  • $\begingroup$ @MiloMoses Then I just don't understand your notation: I thought that $d<Q$ means $d\in[1,2,\dots,Q]$ and $\pi(Q)$ is the usual prime counting function, but, apparently, you mean something else. What is it then? $\endgroup$ – fedja Jan 10 at 9:49
  • $\begingroup$ @fedja The notation you assumed I am using is correct. The bound $Q$ is immediate, I agree, but the bound $\frac{2^{\pi(Q)}}{\sqrt{\pi(Q)}}$ was used instead to illustrate a bound which uses properties of the Mobius function, even if it is less sharp. By "The bound $\pi(Q)$ is, of course not so immediate" I thought you were referring to the bound $\frac{2^{\pi(Q)}}{\sqrt{\pi(Q)}}$ as the "bound based on $\pi(Q)$", but I see now that that was a misunderstanding. $\endgroup$ – Milo Moses Jan 10 at 16:17
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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).

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    $\begingroup$ Thank you SO MUCH for this reference!!!!!!! $\endgroup$ – Milo Moses Jan 10 at 6:08
  • $\begingroup$ Hi, do you know if this is well-known, that $f(x)-\sum_{\Im(\rho)>0} e^{i\rho x}$ is analytic at $0$ for some "elementary" function $f$, it should give non-obvious information on the distributions of non-trivial zeros. Also the expression seems to be continuous in $\epsilon$ when replacing $\zeta(s)$ by $\zeta(s)+\epsilon L(s,\chi)$, which is surprising. $\endgroup$ – reuns Mar 31 at 7:42

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