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

TL;DR

The paper reveals a new high-order correlation between specific functions related to the Riemann zeta function on disconnected sets, leading to novel asymptotic formulas beyond existing theories.

Contribution

It introduces a new correlation analysis between $Z^2[ ho_1(t)]$ and $ ilde{Z}^2(t)$ on disconnected sets, deriving asymptotic formulas not accessible by prior methods.

Findings

Identifies a fourth-order correlation between the functions.

Derives new asymptotic formulas outside known theories.

Highlights the role of Jacob's ladders in nonlocal interactions.

Abstract

It is shown in this paper that there is a fine correlation of the fourth order between the functions $Z^2[\vp_1(t)]$ and $\tilde{Z}^2(t)$, respectively. This correlation is with respect to two collections of disconnected sets. Corresponding new asymptotic formulae cannot be obtained within known theories of Balasubramanian, Heath-Brown and Ivic.