Communications of the
Korean Mathematical Society CKMS

In this paper, we establish an asymptotic formula for mean value of $L^{(k)}(frac{1}{2},chi_{P})$ averaging over $mb P_{2g+1}$ and over $mb P_{2g+2}$ as $g oinfty$ in odd characteristic. We also give an asymptotic formula for mean value of $L^{(k)}(frac{1}{2},chi_{u})$ averaging over $mc I_{g+1}$ and over $mc F_{g+1}$ as $g oinfty$ in even characteristic.

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

MSC numbers: 11M38, 11M06, 11G20, 11M50

Supported by: This work was conducted during the research year of Chungbuk National University in 2021.