Jacob's ladders, conjugate integrals, external mean-values and other properties of a multiply $\pi(T)$-autocorrelation of the function $|\zf|^2$

TL;DR

This paper introduces new transformation formulas for integrals involving products of |z|^2 over disconnected sets on the critical line, advancing understanding of autocorrelations in number theory.

Contribution

It presents a novel class of transformation formulas for integrals of |z|^2 without derivatives, related to autocorrelations of the Riemann zeta function.

Findings

New transformation formulas for integrals involving |z|^2

Insights into autocorrelation properties of the zeta function

Advancement in analysis of disconnected sets on the critical line

Abstract

In this paper we obtain a new class of transformation formulae (without an explicit presence of a derivative) for the integrals containing products of factors $|\zf|^2$ with respect to two components of a disconnected set on the critical line.