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In this paper we introduce the iterations $\phi^k_1(t)$ of the Jacob'sladder. 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 beobtained in known theories of Balasubramanian, Heath-Brown and Ivic.

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  $n\in\mbb{N}$