On Some Algebraic Ways to Calculate Zeros of the Riemann Zeta Function.

The Riemann zeta function is an important number-theoretical tool for studying prime numbers. The first part of the paper is a short survey of some known results about this function. The emphasis is given to the possibility to formulate the celebrated Riemann Hypothesis as a statement from class Π 1 0 in the arithmetical hierarchy. In the second part of the paper the author demonstrates by numerical examples some non-evident ways for finding zeros of the zeta function. Calculations require the knowledge of the value of this function and of N its initial derivatives at one point and consist in solving N systems of linear equations with N unknowns. These methods are not intended for practical calculations but are supposed to be useful for the study of the zeros.