Jacob's ladders, their iterations and the new class of integrals connected with parts of the Hardy-Littlewood integral of the function $|\zeta(1/2+it)|^2$

TL;DR

This paper introduces iterative Jacob's ladders and a new class of integrals involving the Riemann zeta function, providing asymptotic formulas and generalizations of classical results related to the Hardy-Littlewood integral.

Contribution

It presents novel iterative constructions of Jacob's ladders and a new class of integrals linked to the zeta function, extending classical formulas.

Findings

Derived asymptotic formulas for the new integrals

Factorization results for the integrals

Generalizations of Selberg's formulas

Abstract

In this paper we introduce the iterations of the Jacob's ladder and the new type of integral containing certain product of the factors $|\zeta|^2$ corresponding to the components of some disconnected set of the critical line. Next, we obtain an asymptotic formula for this integral, its factorization and, for example, the essential generalization of two Selberg's formulae (1946).