On the Speiser equivalent for the Riemann hypothesis

Abstract

Speiser showed that the Riemann hypothesis is equivalent to the absence of non-trivial zeros of the derivative of the Riemann zeta-function left of the critical line. We investigate the relationship between the non-trivial zeros of the functions belonging to the extended Selberg class and of their derivatives left of the critical line. Every element of this class satisfies a functional equation of the Riemann type, but it contains zeta-functions for which the Riemann hypothesis is not true. As an example, we study the relationship between the trajectories of zeros of a certain linear combination of Dirichlet $$L$$ -functions and of its derivative computationally. In addition, we examine Speiser type equivalent for Dirichlet $$L$$ -functions with imprimitive characters for which the Riemann hypothesis is not true and which do not satisfy the Riemann type functional equation.