Jacob's ladders and the nonlocal interaction of the function $Z(t)$ with the function $\tilde{Z}^2(t)$ on the distance $\sim (1-c)\pi(t)$ for a collection of disconnected sets

TL;DR

This paper reveals a third-order correlation between specific functions related to the Riemann zeta function on disconnected sets, introducing a new asymptotic formula beyond existing theories.

Contribution

It introduces a novel asymptotic formula demonstrating a third-order correlation between $Z[ ho_1(t)]$ and $ ilde{Z}^2(t)$ on disconnected sets, not achievable by prior methods.

Findings

Identifies a third-order correlation between $Z[ ho_1(t)]$ and $ ilde{Z}^2(t)$

Derives a new asymptotic formula for these functions

Shows correlation on collections of disconnected sets

Abstract

It is shown in this paper that there is a fine correlation of the third order between the values of the functions $Z[\vp_1(t)]$ and $\tilde{Z}^2(t)$ which corresponds to two collections of disconnected sets. The corresponding new asymptotic formula cannot be obtained within known theories of Balasubramanian, Heath-Brown and Ivic.