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

Microscopic interpolation formula for the Riemann $Z(t)$-function and  new algorithm for asymptotic solution of some Diophantine equation with  Leibnitz coefficients