Comptes Rendus. Mathématique (Jan 2022)
On the denominators of harmonic numbers. IV
Abstract
Let $\mathcal{L}$ be the set of all positive integers $n$ such that the denominator of $1+1/2+\cdots +1/n$ is less than the least common multiple of $1, 2, \dots , n$. In this paper, under a certain assumption on linear independence, we prove that the set $\mathcal{L}$ has the upper asymptotic density $1$. The assumption follows from Schanuel’s conjecture.
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