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 approximately the same number of non-real zeros left of the critical line. R. Spira showed that ζ ′ (1/2 + it) = 0 implies 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) have the same number of zeros. We prove our result for more general zeta-functions from the extended Selberg class S. We also consider zero trajectories of a certain family of zeta-functions from S.