  In this paper we obtain the first mean-value theorems for the function $Z(t)$
on some disconnected sets. Next, we obtain a geometric law that controls
chaotic behavior of the graph of the function $Z(t)$. This paper is the English
version of the papers \cite{8} and \cite{9}, except of the Appendix that
connects our results with the theory of Jacob's ladders, namely new third-order
formulae have been obtained.
