We give a new construction of an operator acting in a Hilbert space such that the Riemann hypothesis on the zeros of the zeta-function is equivalent to the existence of an eigenvector for this operator with eigenvalue −1. We give also the construction of a dynamical system which turns out to be related to the Riemann hypothesis in the following way: there exists a complex zero of
the zeta-function not lying on the critical line if and only if there is a periodic trajectory of order two having a special form for this dynamical system. The representation of the Riemann zeta-function by means of the infinite product of concrete matrices of order two lies on the basis of this construction.
Mathematical problems and theory of numerical methods