For x irrational, we study the convergence of series of the form $\sum n^{-s}f(nx)$ where f is a real-valued, 1-periodic function which is continuous, except for singularities at the integers with a potential growth. We show that it is possible to fully characterize the convergence set and to approximate the series in terms of the continued fraction of x. This improves and generalizes recent results by Rivoal who studied the examples $f(t)=\cot (\pi t)$ and $f(t)={\sin }^{-2}(\pi t)$.

中文翻译：

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某些丢番图级数的收敛

对于x无理数，我们研究形式级数的收敛性$\sum n^{-秒}F(nX)$其中f是一个实值的 1-周期函数，它是连续的，除了具有潜在增长的整数处的奇点。我们表明可以完全表征收敛集并根据x 的连分数来近似级数。这改进并概括了研究示例的 Rivoal 最近的结果$F(吨)=\mathrm{婴儿床}(\pi 吨)$ 和 $F(吨)={\mathrm{罪}}^{-2}(\pi 吨)$.