Shiu 2016 denominators harmonic numbers revised

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conjecture_p2: States the paper's conjecture that the number of n up to x whose harmonic denominator equals lcm(1, ..., n) lies between two constant multiples of x/log x, with the heuristic behind it and the computation to 10000.

theorem_1: States that consecutive harmonic numerators are never equal and that each of the five order relations between consecutive numerators or denominators occurs infinitely often, with the elementary proofs pointed to.

theorem_2: Characterizes the integers n for which an odd prime p divides lcm(1, ..., n)/d_n through the leading digit of n in base p, the exact criterion behind the trivial half of Problem 291.

theorem_3: States that, for an odd prime p, the set of n with p dividing lcm(1, ..., n)/d_n has harmonic density (1/log p) times the sum of log(1 + 1/m) over the m in E_p, and corrects the asymptotic printed in the counting display (3) of its proof.

theorem_4: States that for odd primes p_1 < ... < p_k whose ratios log p_1/log p_i are linearly independent, some n has p_1 ... p_k dividing lcm(1, ..., n)/d_n, proved by aligning the intervals of Theorem 2 with Kronecker's theorem.

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Peter Shiu, The denominators of harmonic numbers (Revised). arXiv:1607.02863 (2016).

The copy read for this card is arXiv:1607.02863v2 (30 July 2024; the paper is dated 29 July 2024), 8 pages, the later of the two arXiv versions (v1 is of 11 July 2016). No journal version is known: the arXiv listing carries no journal reference and a Crossref bibliographic query on 2026-09-18 found none, so the paper is an unrefereed preprint. Its text layer is clean, and the statements below were checked in it against the PDF pages. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1607.02863), every other right reserved.

Read status: claims checked. Theorems 1--4 and the Conjecture (p. 2) were read clause by clause; the proofs of Theorems 1--3 (pp. 3--4) were read through, the proof of Theorem 4 (pp. 5--6) for structure only, and none is verified here. Result pages exist for Theorems 1--4 and the Conjecture; Lemma 1 (p. 5) serves only the proof of Theorem 4 and has no page.

Writing H_n = c_n/d_n in lowest terms and D_n = LCM(1,...,n), the paper studies how far d_n falls short of D_n. Theorem 1 shows c_n differs from c_{n-1} for all n > 1 and that each of d_n > d_{n-1}, d_n = d_{n-1}, d_n < d_{n-1}, c_n > c_{n-1} and c_n < c_{n-1} occurs infinitely often, so neither sequence is monotonic. With D_n = d_n q_n and Q_p = {n : p | q_n}, Theorem 2 characterizes Q_p by the intervals m p^a <= n < (m+1) p^a with a >= 1 and m in E_p = {n : 1 < n < p, p | c_n}, and Theorem 3 evaluates the harmonic density delta(Q_p) = (1/log p) sum_{m in E_p} log(1 + 1/m). Theorem 4 uses Kronecker's theorem to show that if log p_1/log p_i are linearly independent for odd primes p_1 < ... < p_k then some q_n is divisible by p_1...p_k, i.e. p_1...p_k d_n | D_n for some n. Display (3) in the proof of Theorem 3 (p. 4) prints Q_p(x) ~ |E_p| x/p along x = p^b, while the exact count it displays, |E_p|(p^b - p)/(p - 1), is asymptotic to |E_p| x/(p - 1); Theorem 3 is unaffected. The paper bears on problem 291, whether d_n = D_n (equivalently n lies outside every Q_p) infinitely often: it is conjectured here, not proved, and the paper adds the quantitative conjecture K_1 x/log x < Qtilde(x) < K_2 x/log x for the counting function of {n : d_n = D_n}.

Source: https://arxiv.org/abs/1607.02863.

Bears on. #291: with $\sum_{k\le n}1/k=a_n/L_n$ as on the problem page, $q_n=(a_n,L_n)$; Theorem 2 is the exact leading-digit criterion for $p\mid(a_n,L_n)$ and settles the trivial half (with $m=p-1$, or by [unit_fractions/shiu_2016_denominators_harmonic_numbers_revised/theorem_1|Theorem 1]); the Conjecture is the quantitative form of the open half and the source of the site's $x/\log x$ heuristic; Theorem 3 gives each set $\{n:p\mid(a_n,L_n)\}$, $p$ an odd prime, its harmonic density, and Theorem 4, under its linear-independence hypothesis, gives $n$ with $(a_n,L_n)$ divisible by a prescribed product of distinct odd primes. The paper proves nothing about the open half. #290: the case $a=1$. With $H_n=c_n/d_n$ in lowest terms, $d_n$ is the problem's $v_{1,n}$, and [unit_fractions/shiu_2016_denominators_harmonic_numbers_revised/theorem_1|Theorem 1] (p. 2 of arXiv:1607.02863v2 of 30 July 2024, read on the page image) says $d_n<d_{n-1}$ holds for infinitely many $n$: extending the block $1,\ldots,n-1$ by one term lowers the denominator infinitely often, which answers the problem's existence question for $a=1$ and gives no $b(a)$ for other $a$. Van Doorn's 2024 paper cites this preprint for that case; v1 (11 July 2016) was not compared.

Results to transcribe.

• Theorem 1: c_n != c_{n-1} for all n > 1, and each of d_n > d_{n-1}, d_n = d_{n-1}, d_n < d_{n-1}, c_n > c_{n-1}, c_n < c_{n-1} holds infinitely often.
• Theorem 2: n lies in Q_p = {n : p | q_n} exactly when m p^a <= n < (m+1) p^a for some m in E_p and a >= 1.
• Theorem 3 (page): for an odd prime p, Q_p has harmonic density delta(Q_p) = (1/log p) sum_{m in E_p} log(1 + 1/m).
• Theorem 4 (page): If 2 < p_1 < ... < p_k are primes and log p_1/log p_i (1 <= i <= k) are linearly independent then p_1 p_2 ... p_k divides some q_n, so p_1...p_k d_n | D_n.
• Conjecture: For some positive constants K_1, K_2, the count Qtilde(x) of n <= x with d_n = D_n satisfies K_1 x/log x < Qtilde(x) < K_2 x/log x for all x > 1.

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