We prove the existence of reciprocity formulae for sums of the form where is a piecewise function, featuring an alternating phenomenon not visible in the classical case where . We deduce bounds for these sums in terms of the continued fraction expansion of .
Nous prouvons l’existence de formules de réciprocité pour des sommes de la forme , où est une fonction par morceaux, qui met en évidence un phénomène d’alternance qui n’apparaît pas dans le cas classique où . Nous déduisons des majorations de ces sommes en termes du développement en fraction continue de .
Accepted:
Published online:
Classification: 11L03, 11A55, 11M35
Keywords: Cotangent sum, continued fraction
Author's affiliations:
@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/} }
TY - JOUR TI - Partial sums of the cotangent function JO - Journal de Théorie des Nombres de Bordeaux PY - 2020 DA - 2020/// SP - 217 EP - 230 VL - 32 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1119/ UR - https://doi.org/10.5802/jtnb.1119 DO - 10.5802/jtnb.1119 LA - en ID - JTNB_2020__32_1_217_0 ER -
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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