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

Yochay Jerby

TL;DR

This work reframes Edwards' speculative link between zeros of the core $Z_0(t)=\cos(\theta(t))$ and zeros of the Hardy $Z$-function within a novel variational paradigm built on Spira's high-order sections. By introducing the $A$-parameter space $\mathcal{Z}_N$ and the associated zeros $t_n(r)$, the authors prove that the Riemann Hypothesis is equivalent to the existence of non-colliding, real-zero trajectories along which $t_n(r)$ never meets $t_{n\pm1}(r)$ for all zeros $t_n$. They demonstrate shortcomings of the classical Newton method and the RS formula for this purpose, while showing that a carefully designed non-colliding path in $\mathcal{Z}_{[t_n/2]}$ provides a robust variational route to RH, validated through a detailed $730{,}120$-th zero example. The paper thus recasts RH as a nonlinear optimization problem in a high-dimensional parameter space, with potential discriminant-geometric and machine-learning–driven extensions that could illuminate the zero dynamics of $Z(t)$ beyond traditional analytic techniques.

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 $Z_0(t)=cos(θ(t))$, where $θ(t)$ 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.

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

TL;DR

This work reframes Edwards' speculative link between zeros of the core and zeros of the Hardy -function within a novel variational paradigm built on Spira's high-order sections. By introducing the -parameter space and the associated zeros , the authors prove that the Riemann Hypothesis is equivalent to the existence of non-colliding, real-zero trajectories along which never meets for all zeros . They demonstrate shortcomings of the classical Newton method and the RS formula for this purpose, while showing that a carefully designed non-colliding path in provides a robust variational route to RH, validated through a detailed -th zero example. The paper thus recasts RH as a nonlinear optimization problem in a high-dimensional parameter space, with potential discriminant-geometric and machine-learning–driven extensions that could illuminate the zero dynamics of beyond traditional analytic techniques.

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 , where is Riemann-Siegel theta function, to zeros of , the Hardy -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 to zeros of through paths in a high-dimensional parameter space , recasting the RH as a modern non-linear optimization problem.
Paper Structure (12 sections, 6 theorems, 28 equations, 5 figures, 1 table)

This paper contains 12 sections, 6 theorems, 28 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Riemann-Siegel Formula - Numerics vs. Theory
  3. The Original Formulation of Edward's Speculation on the Riemann-Siegel Formula and Related Questions
  4. The Relation of the Zeros of $Z_0(t)$ and the Zeros of $Z(t)$
  5. The Infeasibility of Tracking the Effect of the Terms of the Riemann-Siegel Equation
  6. The Failure of the Classical Newton Method to Transition from Zeros of $Z_0(t)$ to Zeros of $Z(t)$
  7. Spira's high-order sections vs. the Riemann Siegel Formula
  8. The Variational Approach to Edwards' Speculation
  9. The $A$-variation method - Example for the $730120$-th Zero
  10. The linear curve
  11. A non-colliding curve
  12. Summary and concluding remarks

Key Result

Theorem 1

The Riemann Hypothesis holds if and only if, for any $n \in \mathbb{Z}$, there exists a path $\gamma(r)$ in $\mathcal{Z}_{\left[ \frac{t_n}{2} \right]}$ from $\overline{a}=\overline{0}$ to $\overline{a}=\overline{1}$, along which $t_n$ does not collide with its adjacent zeros $t_{n\pm 1}$. That is,

Figures (5)

  • Figure 1: Graph of $ln \lvert Z(t)\rvert$ (blue) and $ln \lvert Z_0(t)\rvert$ (orange) for $0 \leq t \leq 50$.
  • Figure 2: Graphs of $ln \lvert Z(t)\rvert$ (orange) and the AFE approximation (blue) in the range $412<t<419$
  • Figure 3: Graphs of $\ln \lvert Z(t)\rvert$ (blue), $\ln \lvert Z_0(t)\rvert$ (orange) and the zeros $t_{n-2}$, $t_{n-1}$ and $t_n$ of $Z_0(t)$ for $n=730121$ (purple) in the region $450613.4 \leq t \leq 450614.6$.
  • Figure 4: Graphs of $ln \lvert Z(t)\rvert$ (orange) and Spira's high-order approximation $\ln \left| Z_{N(t)}(t) \right|$ (blue) within the range $412 < t < 419$
  • Figure 5: Schematic illustration of the core $Z_0(t) = Z_N (t ; \overline{0})$ connected to $Z_N(t; \overline{1})$ through two different curves in the $A$-parameter space $\mathcal{Z}_N$.

Theorems & Definitions (16)

  • Conjecture : RH
  • Theorem 1: Edwards' Speculation for High-Order Sections
  • Definition 3.1: The core of the $Z$-function
  • Proposition 3.2: Zeros of the core $Z_0(t)$
  • Remark 4.1: Franca-LeClair
  • Example 6.1: The first Lehmer pair of zeros
  • Example 6.2: Failure of Newton's method for the $730120$-th zero
  • Corollary 6.1: Failure of Newton's method
  • Theorem 7.1: J4
  • Remark 7.2
  • ...and 6 more