Partial sums of the cotangent function

Sandro Bettin; Sary Drappeau

doi:10.5802/jtnb.1119

Sandro Bettin ¹; Sary Drappeau ²

¹ DIMA - Dipartimento di Matematica Via Dodecaneso, 35 16146 Genova, Italy
² Aix Marseille Université, CNRS Centrale Marseille, I2M UMR 7373 13453 Marseille, France

Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 217-230.

Abstract
Résumé

We prove the existence of reciprocity formulae for sums of the form $\sum_{m = 1}^{k - 1} f (\frac{m}{k}) cot (π \frac{m h}{k})$ where $f$ is a piecewise $C^{1}$ function, featuring an alternating phenomenon not visible in the classical case where $f (x) = x$ . We deduce bounds for these sums in terms of the continued fraction expansion of $h / k$ .

Nous prouvons l’existence de formules de réciprocité pour des sommes de la forme $\sum_{m = 1}^{k - 1} f (\frac{m}{k}) cot (π \frac{m h}{k})$ , où $f$ est une fonction $C^{1}$ par morceaux, qui met en évidence un phénomène d’alternance qui n’apparaît pas dans le cas classique où $f (x) = x$ . Nous déduisons des majorations de ces sommes en termes du développement en fraction continue de $h / k$ .

Received: 2019-06-06
Accepted: 2019-07-04
Published online: 2020-08-21

DOI: 10.5802/jtnb.1119
Classification: 11L03, 11A55, 11M35
Keywords: Cotangent sum, continued fraction
Author's affiliations:

Sandro Bettin ¹; Sary Drappeau ²

¹ DIMA - Dipartimento di Matematica Via Dodecaneso, 35 16146 Genova, Italy
² Aix Marseille Université, CNRS Centrale Marseille, I2M UMR 7373 13453 Marseille, France

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Sandro Bettin; Sary Drappeau. Partial sums of the cotangent function. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 217-230. doi : 10.5802/jtnb.1119. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1119/

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