Partial sums of the cotangent function

Bettin, Sandro; Drappeau, Sary

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 ${\sum }_{m=1}^{k-1}f\left(\frac{m}{k}\right)cot\left(\pi \frac{mh}{k}\right)$, 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 $f\left(x\right)=x$. 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 ${\sum }_{m=1}^{k-1}f\left(\frac{m}{k}\right)cot\left(\pi \frac{mh}{k}\right)$ where $f$ is a piecewise $C^1$ function, featuring an alternating phenomenon not visible in the classical case where $f\left(x\right)=x$. We deduce bounds for these sums in terms of the continued fraction expansion of $h/k$.

Classification : 11L03, 11A55, 11M35
Mots clés : Cotangent sum, continued fraction

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