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

TL;DR

This paper investigates the zeros of the derivative of Dirichlet L-functions, removing the possibility of vagrant zeros for large conductors, and refines asymptotic formulas and characterizations related to the generalized Riemann hypothesis.

Contribution

It eliminates vagrant zeros for large conductors and enhances asymptotic zero count formulas, also establishing Speiser-type theorems for derivatives of Dirichlet L-functions.

Findings

Vagrant zeros are excluded for large conductors.

Improved asymptotic formulas for zero counts.

Analogues of Speiser's theorem established.

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.