Partial sums of the cotangent function
Sandro Bettin1; Sary Drappeau2
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.

We prove the existence of reciprocity formulae for sums of the form ∑ m=1 k-1 f(m k)cot(πmh 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 ∑ m=1 k-1 f(m k)cot(πmh 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:
Accepted:
Published online:
DOI: 10.5802/jtnb.1119
Classification: 11L03, 11A55, 11M35
Keywords: Cotangent sum, continued fraction
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
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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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