发表机构
Central South University（中南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究 zeta-star 对应的右导数，证明其连续性、跳跃与维数性质，并给出 Hölder 指数与 Hausdorff 谱的线性结果。
AI 中文摘要
我们研究了二进制展开与无穷深度的多重 zeta-star 值之间的 zeta-star 对应的右导数。在有限指标点，该导数是两个有限深度多重 zeta-star 值的归一化差。我们利用这一恒等式，用低截断的 star 和及其边界系数来表达其局部行为。求和变量三处的齐次贡献以 $2/3$ 的比率收缩；完整系数还接收一个正强迫项。剩余系数的加权和每一步二进制收缩至多 $1/2$。在开参数区间上，我们证明导数处处右连续，并且恰好在非二进点连续。其在二进点的左跳跃由一个显式正多重级数给出，发散点恰好在 $1/2-2^{-M}$，$M\ge2$。指标的细化给出每个内点处的无穷上右 Dini 导数。我们还证明了图的 Hausdorff、packing 和修正上盒维数均为 $\log_2(8/3)$。Hausdorff 下界使用了参数为 $2/3$ 的 Bernoulli 卷积的维数，以及一个独立的测度传递论证。最后，非二进点处的逐点 Hölder 指数为 $\log_2(3/2)$ 除以其二进逼近指数，相应的 Hausdorff 谱是线性的。
英文摘要
We study the right derivative of the zeta-star correspondence between binary expansions and multiple zeta-star values of infinite depth. At a finite-index point, this derivative is a normalized difference of two finite-depth multiple zeta-star values. We use this identity to express its local behavior in terms of lower-truncated star sums and their boundary coefficients. The homogeneous contribution at summation variable three contracts with ratio $2/3$; the full coefficient also receives a positive forcing term. A weighted sum of the remaining coefficients contracts by at most $1/2$ at each binary step. On the open parameter interval, we prove that the derivative is right-continuous everywhere and continuous precisely at the non-dyadic points. Its left jump at a dyadic point is given by an explicit positive multiple series, with divergence exactly at $1/2-2^{-M}$, $M\ge2$. A refinement of the index gives an infinite upper right Dini derivative at every interior point. We also prove that the Hausdorff, packing, and modified upper box dimensions of the graph are all $\log_2(8/3)$. The Hausdorff lower bound uses the dimension of the Bernoulli convolution with parameter $2/3$, together with a separate measure-transfer argument. Finally, the pointwise Hölder exponent at a non-dyadic point is $\log_2(3/2)$ divided by its dyadic approximation exponent, and the corresponding Hausdorff spectrum is linear.
Comments28 pages
