Journal of the
Korean Mathematical Society JKMS

In this series, we investigate the calculation of mean values of derivatives of Dirichlet $L$-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields. The present paper generalizes the results obtained in the first paper. For $\mu\geq1$ an integer, we compute the mean value of the $\mu$-th derivative of quadratic Dirichlet $L$-functions over the rational function field. We obtain the full polynomial in the asymptotic formulae for these mean values where we can see the arithmetic dependence of the lower order terms that appears in the asymptotic expansion.

Keywords: Function fields, derivatives of $L$-functions, moments of $L$-functions, quadratic Dirichlet $L$-functions, random matrix theory

MSC numbers: Primary 11M38; Secondary 11M06, 11G20, 11M50, 14G10

Supported by: The first author is grateful to the Leverhulme Trust (RPG-2017-320) for the support through the research project grant ``Moments of $L$-functions in Function Fields and Random Matrix Theory". The second author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (2020R1F1A1A01066105).