On Edwards' Speculation and a New Variational Method for the Zeros of the $Z$-Function

Abstract

In his foundational book, Edwards introduced a unique "speculation" regarding the possible theoretical origins of the Riemann Hypothesis, based on the properties of the Riemann-Siegel formula. Essentially Edwards asks whether one can find a method to transition from zeros of $Z_0(t)=cos(\theta(t))$, where $\theta(t)$ is Riemann-Siegel theta function, to zeros of $Z(t)$, the Hardy $Z$-function. However, when applied directly to the classical Riemann-Siegel formula, it faces significant obstacles in forming a robust plausibility argument for the Riemann Hypothesis. In a recent work, we introduced an alternative to the Riemann-Siegel formula that utilizes series acceleration techniques. In this paper, we explore Edwards' speculation through the lens of our accelerated approach, which avoids many of the challenges encountered in the classical case. Our approach leads to the description of a novel variational framework for relating zeros of $Z_0(t)$ to zeros of $Z(t)$ through paths in a high-dimensional parameter space $\mathcal{Z}_N$, recasting the RH as a modern non-linear optimization problem.