In Search of Approximate Polynomial Dependencies Among the Derivatives of the Alternating Zeta Function
Yuri Matiyasevich
TL;DR
The paper investigates approximate polynomial dependencies between the Dirichlet eta function $\eta(s)$ and its derivatives, motivated by the known nonexistence of an exact polynomial differential equation for $\eta$ (and by extension $\zeta$). It develops a determinant-based framework with $V_N$ and $W_N$ that predicts $\frac{V_N}{W_N}\to 1$ for generic arguments, and extensively tests this idea via numerical experiments on structured sampling sets, linking the behavior to characteristic polynomials and minor-generalizations. The authors propose a suite of conjectures (covering limiting cases and various modifications) supported by substantial computational evidence, and they demonstrate how shifting indices, omitting derivatives, and using weighted or generalized polynomials still preserve the approximate identity. The findings suggest a robust, near-algebraic structure underlying $\eta(s)$ and its derivatives, offering potential computational tools and a blueprint that might extend to the Riemann zeta function in future work.
Abstract
It is well-known that the Riemann zeta function does not satisfy any exact polynomial differential equation. Here we present numerical evidence for the existence of approximate polynomial dependencies between the values of the alternating zeta function and its initial derivatives. A number of conjectures is stated.