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

TL;DR

This paper explores Edwards' speculation on the origins of the Riemann Hypothesis by using a new variational method based on an accelerated series approach, framing the problem as a nonlinear optimization in a high-dimensional space.

Contribution

It introduces a novel variational framework that connects zeros of $Z_0(t)$ to zeros of $Z(t)$ using an alternative series acceleration technique, avoiding classical obstacles.

Findings

Develops a high-dimensional parameter space for zero analysis

Recasts the Riemann Hypothesis as a nonlinear optimization problem

Provides a new perspective on Edwards' speculation using modern methods

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.