Two estimates on the distribution of zeros of the first derivative of Dirichlet $L$-functions under the generalized Riemann hypothesis

Abstract: Zeros of the Riemann zeta function and its derivatives have been studied by many mathematicians. Among, the number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, Akatsuka, and the author. Berndt, Levinson, and Montgomery investigated the unconditional case, while Akatsuka and the author gave sharper estimates under the truth of the Riemann hypothesis. In this paper, we prove similar results related to the first derivative of the Dirichlet $L$-functions associated with primitive Dirichlet characters under the assumption of the generalized Riemann hypothesis.