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 myself. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka and I gave sharper estimates under the truth of the Riemann hypothesis. Analogous to the case of the Riemann zeta function, the number of zeros and many other properties of zeros of the derivatives of Dirichlet L-functions associated with primitive Dirichlet characters were studied by Yildirim.

 

In this talk, we improve some results shown by Yildirim for the first derivative and show some new results. We also introduce two improved estimates on the distribution of zeros obtained under the truth of the generalized Riemann hypothesis. Finally, we introduce an equivalence condition for the generalized Riemann hypothesis, stated in terms of the distribution of zeros of the first derivative of Dirichlet L-functions associated with primitive Dirichlet characters.