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

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.