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arXiv · 2507.16536

M\"obius inversion and coprime summation for error-sum functions of continued fractions

Min Woong Ahn
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Abstract

We study the unweighted error-sum function $\mathcal{E}(x) \coloneqq \sum_{n \geq 0} ( x- p_n(x)/q_n(x) )$, where $p_n(x)/q_n(x)$ is the $n$th convergent of the continued fraction expansion of $x \in \mathbb{R}$. We prove that the Hausdorff dimension of the graph of $\mathcal{E}$ is exactly equal to $1$. Our proof is number-theoretic in nature and involves M\"obius inversion, summation over coprime convergent denominators, and precise upper bounds derived via continued fraction recurrence relations. As a supplementary result, we rederive the known upper bound of $3/2$ for the Hausdorff dimension of the graph of the relative error-sum function $P(x) \coloneqq \sum_{n \geq 0} (q_n(x)x-p_n(x))$.

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Min Woong Ahn. 2025-07-22. M\"obius inversion and coprime summation for error-sum functions of continued fractions. https://arxiv.org/abs/2507.16536

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