Jacob's ladders and the nonlocal interaction of the function $|\zeta(1/2+it)|$ with the function $\arg\zeta(1/2+it)$ on the distance $\sim(1-c)\pi(t)$

TL;DR

This paper introduces a novel mixed formula linking the magnitude and argument of the Riemann zeta function on the critical line, revealing nonlocal interactions not accessible through classical methods.

Contribution

It presents a new type of formula connecting |(1/2+it)| and  ext{arg}(1/2+it), which cannot be derived from classical theories of Selberg or other known approaches.

Findings

Established a new mixed formula for  and  ext{arg} on the critical line

Revealed nonlocal interactions between || and  ext{arg}

Demonstrated limitations of classical methods in deriving such formulas

Abstract

In this paper we obtain a new-type formula - \emph{a mixed formula} - which connects the functions $|\zeta(1/2+it)|$ and $\arg\zeta(1/2+it)$. This formula cannot be obtained in the classical theory of A. Selberg, and, all the less, in the theories of Balasubramanian, Heath-Brown and Ivic.