# Jacob's ladders, crossbreeding and new synergetic formulas for the class of more complicated external parts of $\zeta$-factorization formulas

## TL;DR

This paper introduces a new canonical synergetic formula involving the Riemann zeta-function and elementary functions, revealing cooperative interactions on the critical line's disconnected sets.

## Contribution

It presents a novel $ ext{ extbackslash zeta}$-analogue of an elementary trigonometric formula, expanding the understanding of zeta-function interactions.

## Key findings

- New $ ext{ extbackslash zeta}$-analogue formula derived
- Reveals cooperative interactions on the critical line
- Applicable to disconnected sets of the zeta-function

## Abstract

In this paper we obtain new canonical synergetic formula, namely an $\zeta$-analogue of next elementary trigonometric formula. This one describes cooperative interactions between corresponding class of elementary functions and the Riemann's zeta-function on a class of disconnected sets on the critical line.