Zeros of the first derivative of Dirichlet $L$-functions

Hirotaka Akatsuka, Ade Irma Suriajaya

Published 2016-04-27Version 1

Y{\i}ld{\i}r{\i}m has classified zeros of the derivatives of Dirichlet $L$-functions into trivial zeros, nontrivial zeros and vagrant zeros. In this paper we remove the possibility of vagrant zeros for the first derivative $L'(s,\chi)$ of Dirichlet $L$-functions when the conductors are large to some extent. Then we improve the error term estimates of asymptotic formulas for the number of zeros of $L'(s,\chi)$. We also establish analogues of Speiser's theorem, which characterize the generalized Riemann hypothesis for $L(s,\chi)$ in terms of zeros of $L'(s,\chi)$ in $0<\operatorname{Re}(s)<1/2$, when the conductor is large.