Hypertranscendence of classical multiple zeta functions

Determine whether classical multiple zeta functions in characteristic 0 are hypertranscendental; equivalently, ascertain whether these functions satisfy any nontrivial algebraic differential equations with polynomial coefficients in their complex variables.

Background

The paper reviews classical results on hypertranscendence: Hölder proved the Euler gamma function is hypertranscendental over the rational function field, Hilbert established the hypertranscendence of the Riemann zeta function, and later work extended such results to broader function classes and base differential fields.

Against this backdrop, the authors note that multiple zeta functions—classical generalizations of the Riemann zeta function in characteristic 0—form a natural next target, but their hypertranscendence status is currently unknown. This contrasts with the paper’s main focus on positive characteristic analogues, where the authors establish algebraic independence and hypertranscendence-type results for deformation series related to gamma and multiple zeta values.

References

There is a generalization of Riemann zeta function, known as the multiple zeta function (see [Mat02] for example). It is natural to ask the hypertranscendence of multiple zeta functions but we do not know that they satisfy or not.

— Hyperderivatives of the deformation series associated with arithmetic gamma values and characteristic $p$ multiple zeta values  (2408.10730 - Harada et al., 2024) in Section 0.2 (Background in characteristic 0)