New consequences of the Riemann-Siegel formula and a law of asymptotic equality of signum-areas of $Z(t)$ function

Abstract: 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.