Does the multiplicative chaos measure arising from random multiplicative functions satisfy a nontrivial modulus of continuity with respect to the Riemann zeta function's critical line behavior, and can its multifractal spectrum be fully characterized in terms of analytic number theoretic data?

The multiplicative chaos measure built from random multiplicative functions is a probabilistic object that encodes deep arithmetic structure. While recent work has established its existence and basic properties, the precise geometric and analytic fine structure of this measure remains unknown. Specifically, we do not know whether the multifractal spectrum of this measure, which describes how the measure concentrates on sets of various Hausdorff dimensions, can be expressed purely in terms of quantities associated with the Riemann zeta function or L-functions, such as moments of the zeta function on the critical line.