Abstract:Recent findings show that the classical Riemann’s non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann’s function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions.

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中文翻译：

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关于黎曼不可微函数的豪斯多夫维数

摘要：最近的研究结果表明，经典黎曼不可微函数具有物理和几何性质，即由副法向流驱动的多边形涡旋丝的不规则轨迹。在本文中，我们给出了其 Hausdorff 维数的上估计。我们还将这个结果适应多重分形设置。为了证明这些结果，我们从新的角度重新计算了黎曼函数围绕有理数的渐近行为，强调了它与 Talbot 效应和高斯和的联系，希望给出其维度的下界并进一步回答是有用的几何问题。

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