Number theory
Small values of signed harmonic sums
Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1062-1074.

For every $τ∈R$ and every integer N, let $mN(τ)$ be the minimum of the distance of τ from the sums $∑n=1Nsn/n$, where $s1,…,sn∈{−1,+1}$. We prove that $mN(τ), for all sufficiently large positive integers N (depending on C and τ), where C is any positive constant less than $1/log⁡4$.

Pour tout $τ∈R$ et tout entier N, soit $mN(τ)$ la distance minimale de τ aux sommes $∑n=1Nsn/n$, où $s1,…,sn∈{−1,+1}$. On montre que $mN(τ) pour tout entier positif N suffisamment grand (dépendant de C et τ), quelle que soit la constante positive C, inférieure à $1/log⁡4$.

Accepted:
Published online:
DOI: 10.1016/j.crma.2018.11.007

Sandro Bettin 1; Giuseppe Molteni 2; Carlo Sanna 3

1 Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy
2 Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy
3 Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
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Sandro Bettin; Giuseppe Molteni; Carlo Sanna. Small values of signed harmonic sums. Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1062-1074. doi : 10.1016/j.crma.2018.11.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.11.007/

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