# Mean values of derivatives of $L$-functions in function fields: IV

## TL;DR

This paper extends previous work on the mean values of derivatives of Dirichlet L-functions in function fields, providing explicit asymptotic formulas that reveal arithmetic dependencies in the lower order terms.

## Contribution

It generalizes earlier results to compute mean values of higher derivatives of quadratic Dirichlet L-functions over function fields, including detailed asymptotic expansions.

## Key findings

- Derived explicit asymptotic formulas for mean values of derivatives
- Identified arithmetic dependence in lower order terms
- Extended previous results to higher derivatives

## Abstract

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.