Partial sums of the cotangent function

Abstract : We prove the existence of reciprocity formulae for sums of the form $\sum_{m=1}^{k-1}f\pr{\frac{m}{k}}\cot\pr{\pi \frac{m h}k}$ 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~$h/k$.
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Sandro Bettin, Sary Drappeau. Partial sums of the cotangent function. Journal de Théorie des Nombres de Bordeaux, Société Arithmétique de Bordeaux, In press. ⟨hal-02189191⟩

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