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

Suriajaya, Ade Irma

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 471-502.

Le nombre de zéros and la distribution des parties réelles des zéros non-réelles de la dérivée de la fonction zêta de Riemann a été étudiée par B. C. Berndt, N. Levinson, H. L. Montgomery, H. Akatsuka et l’auteure. Berndt, Levinson et Montgomery ont étudié le cas inconditionnel, alors qu’Akatsuka et l’auteure ont donné de meilleures estimations sous l’hypothèse de Riemann. Récemment F. Ge a amélioré l’estimation du nombre de zéros par Akatsuka. Dans cet article nous montrons des résultats similaires relatifs à la dérivée des fonctions $L$ de Dirichlet associées aux caractères primitifs de Dirichlet, sous l’hypothèse de Riemann généralisée.

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 B. C. Berndt, N. Levinson, H. L. Montgomery, H. 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. Recently, F. Ge improved the estimate on the number of zeros shown by Akatsuka. In this paper, we prove similar results related to the first derivative of Dirichlet $L$-functions associated with primitive Dirichlet characters under the assumption of the generalized Riemann hypothesis.

Classification : 11M06
Mots clés : Dirichlet $L$-functions, first derivative, zeros

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