  In this paper we use the values of the Riemann $Z(t)$-function in order to
construct certain quasi-orthonormal system of vectors. On this basis we prove a
formula for microscopic interpolation of the function $Z(t)$. Simultaneously we
have obtained a new algorithm how to construct asymptotic solutions of
Diophantine equation with Leibnitz coefficients. This paper is English version
of the paper \cite{3} except the Appendix.
