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Source. Theorem 2 and Conjecture 1, Section 1, printed p. 54 (PDF p. 3) of the retained publisher's PDF, C. R. Math. Acad. Sci. Paris 360 (2022), 53--57, DOI 10.5802/crmath.282; proof in Section 3, printed pp. 55--57. Read on the PDF pages in the text layer. Refereed journal article (received 11 August 2021, accepted 12 October 2021, published online 26 January 2022, as printed on p. 53 and in the Crossref record read). Notation: $H_n=u_n/v_n$ in lowest terms; $\mathcal L$ is the set of $n\ge1$ with $v_n<\mathrm{lcm}(1,\ldots,n)$; $\bar d(\mathcal L)=\limsup_{x\to\infty}\mathcal L(x)/x$.

Statement

Conjecture 1. If $q_1,\ldots,q_l$ are distinct primes, then $1/\log q_1,\ldots,1/\log q_l$ are linearly independent over $\mathbb Q$. (The paper derives it from the weak Schanuel conjecture: for multiplicatively independent nonzero algebraic $\beta_1,\ldots,\beta_m$ the numbers $\log\beta_i$ are algebraically independent.)

Theorem 2. Assuming Conjecture 1, $\bar d(\mathcal L)=1$.

Proof pointer and sketch (Section 3)

Let $p_i$ be the $i$th prime and $a_i=\prod_{2\le j\le i}(1-1/p_j)$, so $a_i\to0$ by Mertens' theorem (Lemma 3); fix $k$ with $a_k<\varepsilon/2$. Conjecture 1 makes $\log p_2/\log p_i$ ($2\le i\le k$) linearly independent over $\mathbb Q$, so Kronecker's theorem (Lemma 4) gives infinitely many $q$ and exponents $s_i$ with $p_i^{s_i}$ within a factor $p_i^{\pm\delta}$ of $a_{i-1}p_2^{\,q}$ (display (2)). For $n$ in $((p_i-1)p_i^{s_i-1},p_i^{s_i})$ the terms of $H_n$ with denominators divisible by $p_i^{s_i-1}$ contribute $H_{p_i-1}/p_i^{s_i-1}$, whose numerator is a multiple of $p_i$, so $v_{p_i}(H_n)\ge-(s_i-2)$ while $p_i^{s_i-1}\mid\mathrm{lcm}(1,\ldots,n)$; hence these intervals lie in $\mathcal L$ (display (3)). Lemma 5 bounds the part of each $(a_ip_2^{\,q},a_{i-1}p_2^{\,q})$ outside the corresponding interval, and summing over $i$ gives $\mathcal L(p_2^{\,q})\ge p_2^{\,q}-\varepsilon p_2^{\,q}-3k$, so $\bar d(\mathcal L)\ge1-\varepsilon$. The two-page proof was read through here, not verified.

Dependencies and read depth

Mertens' theorem and Kronecker's theorem (Hardy and Wright, Theorems 429 and 442), plus Conjecture 1 as an explicit hypothesis. Read depth: claims checked; the proof read through, not verified; nothing independently reviewed.

Relation to Problem 291

With $\sum_{k\le n}1/k=a_n/L_n$ and $L_n=\mathrm{lcm}(1,\ldots,n)$, $v_n=L_n/(a_n,L_n)$, so $n\in\mathcal L$ exactly when $(a_n,L_n)>1$. Theorem 2 says, conditionally, that this half of Problem 291 holds on a set of upper density $1$; unconditionally that half is settled by the leading-digit criterion (a set of positive lower density). The theorem says nothing about the other half, $(a_n,L_n)=1$ infinitely often, and Conjecture 1 remains open (a 2011 MathOverflow question asking for it had no answer on 2026-09-18).

Bears on. #291 (conditional density statement for the trivial half).