Zeros of the extended Selberg class zeta-functions and of their derivatives

Abstract

Levinson and Montgomery proved that the Riemann zeta-function ζ(s) and its derivative have approximatelythe same number of nonreal zeros left of the critical line. Spira showed that ζ′(1/2+it) = 0 implies that ζ(1/2+it) = 0.Here we obtain that in small areas located to the left of the critical line and near it the functions ζ(s) and ζ′(s) havethe same number of zeros. We prove our result for more general zeta-functions from the extended Selberg class S . Wealso consider zero trajectories of a certain family of zeta-functions from S .