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

TL;DR

This paper investigates high-order sections of the Hardy Z-function, providing theoretical support for Spira's conjecture that a certain approximation satisfies the Riemann Hypothesis, linking the approximation's zeros to RH.

Contribution

The paper offers a theoretical justification for Spira's conjecture, connecting the approximation of the Hardy Z-function to the Riemann Hypothesis using new series acceleration techniques.

Findings

Spira's approximation likely satisfies RH, with all zeros real.

New series acceleration methods are developed for analyzing Z-function approximations.

The approximation's properties are shown to be essentially equivalent to RH.

Abstract

Sections of the Hardy $Z$-function are given by $Z_N(t) := \sum_{k=1}^{N} \frac{cos(\theta(t)-ln(k) t) }{\sqrt{k}}$ 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 $\widetilde{N}(t) = \left [ \sqrt{\frac{t}{2 \pi}} \right ]$. (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.