Jacob's ladders and vector operator producing new generations of $L_2$-orthogonal systems connected with the Riemann's $\zeta(\frac 12+it)$ function

TL;DR

This paper introduces a novel vector-operator that generates infinite new $L_2$-orthogonal systems linked to the Riemann zeta-function and Jacob's ladder, expanding the functional analysis framework in number theory.

Contribution

The paper presents a new generating vector-operator that produces infinite $L_2$-systems connected with the Riemann zeta-function and Jacob's ladder, offering a fresh approach to orthogonal systems.

Findings

Generated new $L_2$-systems depending on the zeta-function

Established connection between Jacob's ladders and orthogonal systems

Provided a method to produce infinite orthogonal systems

Abstract

In this paper we introduce a generating vector-operator acting on the class of functions $L_2([a,a+2l])$. This operator produces (for arbitrarily fixed $[a,a+2l]$) infinite number of new generation $L_2$-systems. Every element of the mentioned systems depends on Riemann's zeta-function and on Jacob's ladder.