Jacob's ladders, new properties of the function $\arg\zf$ and corresponding metamorphoses

TL;DR

This paper introduces new mathematical properties and interactions of the argument of the Riemann zeta function using Jacob's ladders, revealing metamorphoses in oscillating systems and advancing understanding of zeta function behavior.

Contribution

It presents novel formulae and metamorphoses related to the argument of the zeta function, utilizing Jacob's ladders and related integrals, which are new contributions to analytic number theory.

Findings

New formulae for interaction of |z| and \argz

Metamorphoses of oscillating Q-system identified

Enhanced understanding of \\argz properties

Abstract

The notion of the Jacob's ladders, reversely iterated integrals and the $\zeta$-factorization is used in this paper in order to obtain new results in study of the function $\arg\zf$. Namely, we obtain new formulae for non-local and non-linear interaction of the functions $|\zf|$ and $\arg\zf$, and also a set of metamorphoses of the oscillating Q-system.