Jacob's ladders, interactions between $\zeta$-oscillating systems and $\zeta$-analogue of an elementary trigonometric identity

TL;DR

This paper introduces a zeta-analogue of a basic trigonometric identity and explores interactions between oscillating systems within the framework of Jacob's ladders and the Riemann zeta function.

Contribution

It presents a novel zeta-analogue of a fundamental trigonometric identity and investigates new interactions between oscillating systems related to the Riemann zeta function.

Findings

Derived a zeta-analogue of an elementary trigonometric identity

Established interactions between oscillating systems

Extended the theory of Jacob's ladders and zeta-factorization

Abstract

In our previous papers, we have introduced within the theory of the Riemann zeta function the following notions: Jacob's ladders, oscillating systems, $\zeta$-factorization, metamorphoses, \dots In this paper we obtain $\zeta$-analogue of an elementary trigonometric identity and other interactions between oscillating systems.