Partial sums of the cotangent function

Sandro Bettin; Sary Drappeau

doi:10.5802/jtnb.1119

Sandro Bettin ; Sary Drappeau

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

Résumé
Abstract

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 $h / k$ .

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 $h / k$ .

Reçu le : 2019-06-06
Accepté le : 2019-07-04
Publié le : 2020-08-21

DOI : https://doi.org/10.5802/jtnb.1119
Classification : 11L03, 11A55, 11M35
Mots clés : Cotangent sum, continued fraction

@article{JTNB_2020__32_1_217_0,
     author = {Sandro Bettin and Sary Drappeau},
     title = {Partial sums of the cotangent function},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {217--230},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {1},
     year = {2020},
     doi = {10.5802/jtnb.1119},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1119/}
}

Sandro Bettin; Sary Drappeau. Partial sums of the cotangent function. Journal de Théorie des Nombres de Bordeaux, Tome 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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