Wu 2022 denominators harmonic numbers iv

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theorem_2: States the conditional theorem that, if the reciprocals of the logarithms of distinct primes are linearly independent over the rationals, the integers whose harmonic denominator falls short of lcm(1, ..., n) have upper asymptotic density one.

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Bing-Ling Wu and Xiao-Hui Yan, On the denominators of harmonic numbers. IV, C. R. Math. Acad. Sci. Paris 360 (2022), 53--57; DOI 10.5802/crmath.282. Published online 26 January 2022; received 11 August 2021, revised 25 August and 18 September 2021, accepted 12 October 2021.

The retained folder-name PDF is the publisher's PDF (Centre Mersenne): a cover page followed by the printed pp. 53--57, so physical PDF p. $n$ is printed p. $51+n$. Its text layer is clean, and the statements below were read in it against the PDF. Provenance: retained from the repository's survey download set of September 2026, identified by its DOI (https://doi.org/10.5802/crmath.282), which the PDF itself names together with the journal site; the download URL was not recorded; 601,771 bytes. The file prints on its cover page (PDF p. 1) "This article is licensed under the Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/", the Creative Commons Attribution 4.0 license.

Read status: claims checked. Conjecture 1 and Theorem 2 were read clause by clause; the two-page proof of Theorem 2 (pp. 55--57) was read through but not checked.

Contents

Setup (p. 53): $H_n=1+1/2+\cdots+1/n=u_n/v_n$ in lowest terms. The introduction recalls Shiu's result that $v_n=v_{n+1}$ for infinitely many $n$ (the paper's [6]), Wu and Chen's result that $v_n=v_{n+1}$ holds on a set of asymptotic density one ([9]), and the bounds on the set $J_p$ of $n$ with $p\mid u_n$ (Eswarathasan--Levine, Sanna, Wu--Chen).

• Definition (p. 54): $\mathcal L$ is the set of positive integers $n$ with $v_n$ less than the least common multiple of $1,\dots,n$ (of which $v_n$ is always a divisor); $\bar d(\mathcal L)=\limsup\mathcal L(x)/x$.
• Weak Schanuel's Conjecture (p. 54): "If $\beta_1,\ldots,\beta_m$ are non-zero, multiplicatively independent algebraic numbers, then $\log\beta_1,\ldots,\log\beta_m$ are algebraically independent." Conjecture 1 (p. 54): for any distinct primes $q_1,\dots,q_l$ the numbers $1/\log q_1,\dots,1/\log q_l$ are linearly independent over $\mathbb Q$; the paper notes it follows from the weak Schanuel conjecture.
• Theorem 2 (p. 54): assuming Conjecture 1, $\bar d(\mathcal L)=1$.
• Proof outline (pp. 54--57): Mertens' theorem (Lemma 3) and Kronecker's theorem (Lemma 4, applied to $\log p_2/\log p_i$ for $i=2,\dots,k$, which Conjecture 1 makes linearly independent) produce infinitely many $q$ and exponents $s_i$ with $p_i^{s_i}$ close to $a_{i-1}p_2^{\,q}$, where $a_i=\prod_{2\le j\le i}(1-1/p_j)$; for $n$ in $((p_i-1)p_i^{s_i-1},p_i^{s_i})$ the $p_i$-adic valuation of $H_n$ is at least $-(s_i-2)$ while $p_i^{s_i-1}$ divides $\mathrm{lcm}(1,\dots,n)$, so these intervals lie in $\mathcal L$ (the paper's (3)); Lemma 5 bounds the part of $(a_ip_2^{\,q},a_{i-1}p_2^{\,q})$ outside the corresponding interval and gives $\mathcal L(p_2^{\,q})\ge(1-\varepsilon)p_2^{\,q}-3k$.

Compiled scope

The statements above were checked in the text layer; the proof was read as an outline only and is not verified here. Nothing here is independently reviewed.

Bears on. #291: writing $H_n=a_n/L_n$ with $L_n=\mathrm{lcm}(1,\dots,n)$ gives $v_n=L_n/(a_n,L_n)$, so $n\in\mathcal L$ exactly when $(a_n,L_n)>1$; Theorem 2 shows, conditionally on Conjecture 1, that this half of the problem holds on a set of upper density $1$, hence for infinitely many $n$; the paper says nothing about the other half, $(a_n,L_n)=1$ for infinitely many $n$.