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

Authors : Ramunas Garunkstis

Pages : 2921-2930

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Publication Date : 9999-12-31

Article Type : Makaleler

Abstract :Levinson and Montgomery proved that the Riemann zeta-function ζ(s) and its derivative have approximately the 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) 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.
Keywords : Riemann zeta-function, extended Selberg class, nontrivial zeros, Speiser's equivalent for the Riemann hypothesis