Mean values of derivatives of quadratic prime Dirichlet $L$-functions in function fields

Commun. Korean Math. Soc. Published online January 11, 2022

Hwanyup Jung
Chungbuk National University

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

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