The $A$-philosophy for the Hardy $Z$-Function

TL;DR

We address the Hardy $Z$-function zeros and the Riemann Hypothesis (RH) by introducing the $A$-variation space $\mathcal{Z}_N$ and local Gram discriminants $\Delta_n(\overline{a})$. The main result proves that RH is equivalent to the corrected Gram's law $(-1)^n \Delta_n(\overline{1})>0$ for all integers $n$, reducing Gram's law to a first-order approximation $Z(g_n;\overline{a})$ and revealing a second-order Hessian structure tied to shifts of Gram points. The work uncovers a dynamic repulsion phenomenon $|Z'(g_n)|>4|Z(g_n)|$ for isolated bad Gram points, and shows how non-linear paths in $\mathcal{Z}_N$ can bypass collisions, offering a novel mechanism toward RH and linking to Montgomery pair correlations. Overall, the discriminant-based, dynamical viewpoint yields new insights into Gram's law, the pair-correlation conjecture, and RH, with potential implications for analytic number theory.

Abstract

In recent works we have introduced the parameter space $\mathcal{Z}_N$ of $A$-variations of the Hardy $Z$-function, $Z(t)$, whose elements are functions of the form \begin{equation} \label{eq:Z-sections} Z_N(t ; \overline{a} ) = \cos(θ(t))+ \sum_{k=1}^{N} \frac{a_k}{\sqrt{k+1} } \cos ( θ(t) - \ln(k+1) t), \end{equation} where $\overline{a} = (a_1,...,a_N) \in \mathbb{R}^N$. The \( A \)-philosophy advocates that studying the discriminant hypersurface forming within such parameter spaces, often reveals essential insights about the original mathematical object and its zeros. In this paper we apply the $A$-philosophy to our space $\mathcal{Z}_N$ by introducing \( Δ_n(\overline{a} ) \) the $n$-th Gram discriminant of \( Z(t) \). We show that the Riemann Hypothesis (RH) is equivalent to the corrected Gram's law \[ (-1)^n Δ_n(\overline{1}) > 0, \] for any $n \in \mathbb{Z}$. We further show that the classical Gram's law \( (-1)^n Z(g_n) >0\) can be considered as a first-order approximation of our corrected law. The second-order approximation of $Δ_n (\overline{a})$ is then shown to be related to shifts of Gram points along the \( t \)-axis. This leads to the discovery of a new, previously unobserved, repulsion phenomena \[ \left| Z'(g_n) \right| > 4 \left| Z(g_n) \right|, \] for bad Gram points $g_n$ whose consecutive neighbours $g_{n \pm 1}$ are good. Our analysis of the \(A\)-variation space \(\mathcal{Z}_N\) introduces a wealth of new results on the zeros of \(Z(t)\), casting new light on classical questions such as Gram's law, the Montgomery pair-correlation conjecture, and the RH, and also unveils previously unknown fundamental properties.

Figures (9)

• Figure 1: Schematic presentation of questions regarding zeros of $Z(t)$: Riemann hypothesis, Gram's law and Montgomery's pair correlation conjecture.
• Figure 2: $\ln \left | Z(t) \right |$ (blue) and $\ln \left | Z_0 (t) ) \right |$ (orange) for $0 \leq t \leq 50$.
• Figure 3: $\ln \left |Z_1(t; a) \right |$ in the range $66.5 \leq t \leq 70$ for $r=0$ (blue), $r=0.75$ (orange), $r=1.5$ (green) and $r=2.25$ (red).
• Figure 4: Graph of $\Delta_n (r)$ (blue) and $Z_N (g_n ; r)$ (orange) for $n=90$ (left) and $n=126$ (right) with $0 \leq r \leq 1$.
• Figure 5: Graphs of $\ln \left |Z_N(t;r) \right |$ in the range $t \in [g_n -2,g_n+2]$ for $n=90$ (left) and $n=126$ (right) and various values of $r \in [0,1]$.

Theorems & Definitions (28)

• Definition : The $n$-th Gram discriminant
• Theorem A: Corrected Gram's law equivalent to RH
• Theorem B: First-order approximation of $\Delta_n(\overline{a})$
• Theorem C: Second-order Hessian of $\Delta_n(r)$
• Theorem D
• Conjecture 1.1: Repulsion for isolated bad Gram points
• Definition 2.2: Core Function
• Theorem 2.1: J4
• Definition 2.4: $N$-th $A$-Parameter Space
• Theorem 2.2: Edwards' Speculation for High-Order Sections J5