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

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(\theta(t))+ \sum_{k=1}^{N} \frac{a_k}{\sqrt{k+1} } \cos ( \theta (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 \( \Delta_n(\overline{a} ) \) the $n$-th Gram discriminant of \( Z(t) \). We show that the Riemann Hypothesis (RH) is equivalent to the corrected Gram's law \[ (-1)^n \Delta_n(\overline{1}) > 0, \] 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 $\Delta_n (\overline{a})$ 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.