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

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 $P(x) \coloneqq \sum_{n \geq 0} (q_n(x)x-p_n(x))$.