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

Source: [Zeros of the first derivative of Dirichlet $L$-functions](<https://www.sota2.com/research/paper/68fa587a1e209131ce9aeb4f>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T15:21:36.715Z

Authors: Hirotaka Akatsuka, Ade Irma Suriajaya

Published: 2016-04-27T10:51:45.000Z

## 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1604.08015>)

[PDF](<https://arxiv.org/pdf/1604.08015.pdf>)