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 $n\in\mbb{N}$

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.