Partial sums of the cotangent function
Journal de Théorie des Nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 217-230.

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.

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.

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Accepté le :
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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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