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

Abstract

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 $L'(s,\chi)$ when the conductors are large to some extent. Then we improve asymptotic formulas for the number of zeros of $L'(s,\chi)$ in $\{s\in\mathbb{C}:\operatorname{Re}(s)>0, |\operatorname{Im}(s)|\leq T\}$. We also establish analogues of Speiser's theorem, which characterize the generalized Riemann hypothesis for $L(s,\chi)$ in terms of zeros of $L'(s,\chi)$, when the conductor is large.