Jacob's ladders, the iterations of Jacob's ladder $\phi^k_1(t)$ and asymptotic formulae for the integrals of the products ... for arbitrary fixed $n\in\mbb{N}$

TL;DR

This paper introduces the iterations of Jacob's ladder functions and establishes asymptotic formulas for integrals involving products of these iterations, revealing new insights beyond existing theories.

Contribution

It presents the concept of iterated Jacob's ladder functions and derives novel asymptotic formulas for their product integrals, extending previous results.

Findings

Mean-value of product asymptotically equals ln^{n+1} T

New asymptotic formulas for iterated Jacob's ladder integrals

Results extend beyond known theories for n=1

Abstract

In this paper we introduce the iterations $\phi^k_1(t)$ of the Jacob's ladder. It is proved, for example, that the mean-value of the product $$Z^2[\phi^n_1(t)]Z^2[\phi^{n-1}(t)]... Z^2[\phi^0_1(t)]$$ over the segment $[T,T+U]$ is asymptotically equal to $\ln^{n+1}T$. Nor the case $n=1$ cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.