Small values of signed harmonic sums

Bettin, Sandro; Molteni, Giuseppe; Sanna, Carlo

doi:10.1016/j.crma.2018.11.007

Number theory

Small values of signed harmonic sums

Presented by: the Editorial Board

Bettin, Sandro ¹; Molteni, Giuseppe ²; Sanna, Carlo 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

Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1062-1074.

Abstract
Résumé

For every $τ \in R$ and every integer N, let $m_{N} (τ)$ be the minimum of the distance of τ from the sums $\sum_{n = 1}^{N} s_{n} / n$ , where $s_{1}, \dots, s_{n} \in {- 1, + 1}$ . We prove that $m_{N} (τ) < \exp (- C {(\log N)}^{2})$ , 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 $τ \in R$ et tout entier N, soit $m_{N} (τ)$ la distance minimale de τ aux sommes $\sum_{n = 1}^{N} s_{n} / n$ , où $s_{1}, \dots, s_{n} \in {- 1, + 1}$ . On montre que $m_{N} (τ) < \exp (- C {(\log N)}^{2})$ pour tout entier positif N suffisamment grand (dépendant de C et τ), quelle que soit la constante positive C, inférieure à $1 / \log 4$ .

Received: 2018-06-23
Accepted: 2018-11-08
Published online: 2018-11-01

DOI: 10.1016/j.crma.2018.11.007

Author's affiliations:

Bettin, Sandro ¹; Molteni, Giuseppe ²; Sanna, Carlo ³

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     title = {Small values of signed harmonic sums},
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Bettin, Sandro; Molteni, Giuseppe; Sanna, Carlo. 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. http://www.numdam.org/articles/10.1016/j.crma.2018.11.007/

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