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On the approximation of the Hardy $Z$-function via high-order sections

Yochay Jerby

TL;DR

The paper analyzes two approaches to approximate the Hardy $Z$-function: the classical Hardy-Littlewood $2Z_{ ilde N(t)}(t)$ with $ ilde N(t)=igl[ rac{t}{2 ilde ext{ } olinebreak ext{ } rac{ } olinebreak ext{ }igr]$ and Spira's high-range section $Z_{N(t)}(t)$ with $N(t)=[t/2]$. It introduces an accelerated AFE based on the Hasse–Sondow expansion, giving $Z(t)= ilde Z_{N(t)}(t)+O(e^{-oldsymbol{ extomega} t})$ where $ ilde Z_{N(t)}(t)$ uses accelerated coefficients $ ildeeta^{acc}_k(t)$, and shows these coefficients converge to Spira's step-coefficients in the $oldsymbol{eta}$-triangle framework. The authors prove that Spira's section zeros correspond to RH zeros of $Z(t)$ by establishing asymptotic equivalence between $Z_{N(t)}(t)$ and the accelerated sections $ ilde Z_{N(t)}(t)$, with a key result $ig\| ildeeta_k^{acc}(t)-eta_k^{step}(t)ig r_{ ext{2}} o 0$ as $t o\infty$. This provides a theoretical justification for Spira's conjecture and suggests a broader parametrized space of sections $Z(t;oldsymbol{eta_k})$ for studying zeros under $oldsymbol{eta_k}\inoldsymbol{ extell{2}}$. The work thus connects accelerated series techniques to RH through the behavior of high-order sections.

Abstract

Sections of the Hardy $Z$-function are given by $Z_N(t) := \sum_{k=1}^{N} \frac{cos(θ(t)-ln(k) t) }{\sqrt{k}}$ for any $N \in \mathbb{N}$. Sections approximate the Hardy $Z$-function in two ways: (a) $2Z_{\widetilde{N}(t)}(t)$ is the Hardy-Littlewood approximate functional equation (AFE) approximation for $\widetilde{N}(t) = \left [ \sqrt{\frac{t}{2 π}} \right ]$. (b) $Z_{N(t)}(t)$ is Spira's approximation for $N(t) = \left [\frac{t}{2} \right ]$. Spira conjectured, based on experimental observations, that, contrary to the classical approximation $(a)$, approximation (b) satisfies the Riemann Hypothesis (RH) in the sense that all of its zeros are real. We present theoretical justification for Spira's conjecture, via new techniques of acceleration of series, showing that it is essentially equivalent to RH itself.

On the approximation of the Hardy $Z$-function via high-order sections

TL;DR

The paper analyzes two approaches to approximate the Hardy -function: the classical Hardy-Littlewood with and Spira's high-range section with . It introduces an accelerated AFE based on the Hasse–Sondow expansion, giving where uses accelerated coefficients , and shows these coefficients converge to Spira's step-coefficients in the -triangle framework. The authors prove that Spira's section zeros correspond to RH zeros of by establishing asymptotic equivalence between and the accelerated sections , with a key result as . This provides a theoretical justification for Spira's conjecture and suggests a broader parametrized space of sections for studying zeros under . The work thus connects accelerated series techniques to RH through the behavior of high-order sections.

Abstract

Sections of the Hardy -function are given by for any . Sections approximate the Hardy -function in two ways: (a) is the Hardy-Littlewood approximate functional equation (AFE) approximation for . (b) is Spira's approximation for . Spira conjectured, based on experimental observations, that, contrary to the classical approximation , approximation (b) satisfies the Riemann Hypothesis (RH) in the sense that all of its zeros are real. We present theoretical justification for Spira's conjecture, via new techniques of acceleration of series, showing that it is essentially equivalent to RH itself.
Paper Structure (7 sections, 2 theorems, 23 equations, 5 figures)

This paper contains 7 sections, 2 theorems, 23 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Riemann's Analytic Extension of Zeta
  3. The Approximate Functional Equation (AFE) and the Riemann-Siegel Formula
  4. Sections of the $Z$-function and Spira's Approximation
  5. Spira's conjecture and the Absence of Theoretical Justification
  6. Spira's Approximation and Accelerated Approximations
  7. Discussion and Conclusions

Key Result

Proposition 2.1

The following holds where is the accelerated $N$-th section with the coefficients $\alpha^{acc}_k(t)$ given by

Figures (5)

  • Figure 1: Values of the sections $Z_N(t)$ for the fixed value of $t=3000$ and $N=1,...,1500$ (blue), the Hardy-Littlewood approximation of $\frac{1}{2}Z(3000)$ (green) and the Spira approximation $Z(3000)$ (brown).
  • Figure 2: Graphs of $ln \lvert Z(t)\rvert$ (orange) and the Spira approximation $ln \lvert Z_N(t)\rvert$ with $N=N(t)=205$ (blue - left) and the classical Hardy-Littlewood approximation $ln \lvert2 Z_N(t)\rvert$ with $N=\widetilde{N}(t)=8$ (blue - right) in the range $412<t<419$
  • Figure 3: Graphs of $ln \lvert Z(t)\rvert$ (orange) and our accelerated approximation $ln \lvert \widetilde{Z}_{N(t)}(t)\rvert$ of J with $N=N(t)=205$ (blue) in the range $412<t<419$.
  • Figure 4: The triangle of coefficients $\beta_{n,k}(t)$ with $A(n,t)$ arising from the horizontal order of summation (red) and $\widetilde{\alpha}_k(t)$ arising from the vertical order of summation (blue).
  • Figure 5: The accelerated coefficients $\widetilde{\alpha}^{acc}_k(t)$ (blue) and the step coefficients $\alpha^{step}_k(t)$ (orange) of Spira's section for $t=400$ and $k=1,...,400$.

Theorems & Definitions (4)

  • Conjecture : Spira's RH for sections
  • Proposition 2.1
  • proof
  • Corollary 2.2