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Source. Theorem 3, Section 1, p. 2 of Peter Shiu, The denominators of harmonic numbers (Revised), arXiv:1607.02863v2 (30 July 2024; the paper is dated 29 July 2024); proof in Section 4, p. 4. Preprint, not published in a journal (arXiv listing checked). Notation: $H_n=c_n/d_n$ in lowest terms, $D_n=\mathrm{lcm}(1,\ldots,n)=d_nq_n$, and for an odd prime $p$, $E_p=\{n:1<n<p,\ p\mid c_n\}$ and $Q_p=\{n:p\mid q_n\}$.

Statement

Theorem 3 (p. 2). For every odd prime $p$, the set $Q_p$ has a harmonic density, and

$$\delta(Q_p)=\frac{1}{\log p}\sum_{m\in E_p}\log\Bigl(1+\frac1m\Bigr).$$

The harmonic density is the one the proof computes (p. 4): $\delta(Q_p)=\lim_{x\to\infty}(\log x)^{-1}\sum_{n<x,\,n\in Q_p}1/n$. Since $p-1\in E_p$ always (p. 2), $\delta(Q_p)\ge\log(p/(p-1))/\log p>0$. For $p=3$, $E_3=\{2\}$ and $\delta(Q_3)=\log(3/2)/\log3$.

Proof pointer and sketch (Section 4)

By Theorem 2, $Q_p$ is the disjoint union of the intervals $mp^a\le n<(m+1)p^a$, $m\in E_p$, $a\ge1$. Each such interval contributes $\log(1+1/m)+O(1/(mp^a))$ to the harmonic sum, so summing over $a<b$ gives $b\sum_{m\in E_p}\log(1+1/m)+O(\log p)$, and the tail between $x$ and the next power of $p$ contributes $O(\log p)$; dividing by $\log x$ gives the limit. The argument is half a page, read through here and not independently reviewed.

The same section counts $Q_p$ along powers of $p$ (display (3), p. 4): for $x=p^b$, the intervals with $a\le b-1$ give $Q_p(x)=|E_p|(p^b-p)/(p-1)$. The print then writes $Q_p(x)\sim|E_p|x/p$ as $x=p^b\to\infty$; the exact count it displays is asymptotic to $|E_p|x/(p-1)$ instead. Recomputed here for $p=3$, $x=3^8$: $Q_3(x)=3279=(3^8-3)/2$, against $x/3=2187$. The discrepancy is in the asymptotic only and does not touch Theorem 3. The paper also remarks (p. 4) that, because $Q_p$ consists of long runs of consecutive integers, it has no asymptotic density.

Dependencies and read depth

Theorem 2 and the elementary estimate $\sum_{x\le n<y}1/n=\log(y/x)+O(1/x)$. Read depth: claims checked; the proof read through, not verified.

Relation to Problem 291

In the notation of Problem 291, $q_n=(a_n,L_n)$, so $Q_p$ is the set of $n$ with $p\mid(a_n,L_n)$. The theorem measures, prime by prime, how much of the integers the second question's answer covers; it is unconditional but says nothing about how the sets $Q_p$ for different $p$ overlap. The paper's Conjecture for the first question rests on display (3) of this proof (Section 7, p. 6), not on the theorem itself.

Bears on. #291 (the harmonic density of each set $\{n:p\mid(a_n,L_n)\}$, $p$ an odd prime; nothing on the first question).