It is obtained results related to the Riemann function ξ(s) which give a new necessary condition for the validity of Riemann hypothesis on the zeros of the classical zeta-function. It is proved that if at least one even derivative of the function ξ(s) at the point s = 1/2 is not positive, then the Riemann hypothesis would be false. However, it is also proved that all the even derivatives at the point s = 1/2 are strictly positive and their asymptotic form for the order of derivatives tending to ininity is found.

Mathematical problems and theory of numerical methods