The derivative function of the zeta-star correspondence

Abstract

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 H\"older exponent at a non-dyadic point is $\log_2(3/2)$ divided by its dyadic approximation exponent, and the corresponding Hausdorff spectrum is linear.