Moments of Higher derivatives of the logarithmic derivative of Dirichlet $L$-functions

Let $\chi$ be a non-principal Dirichlet character and $L(s,\chi )$ be the associated Dirichlet $L$-function. Let us use $\mathcal{L}(s,\chi )$ to denote its logarithmic derivative $L^{\mathrm{\prime }}(s,\chi )\mathrm{/}L(s,\chi )$. We first prove some arithmetic formulas for higher derivatives ${\mathcal{L}}^{(r)}(1,\chi )$. We then investigate their moments. We study the average of $P^{(a,b)}({\mathcal{L}}^{(r)}(1,\chi ))$ as $\chi$ runs over all non-principal Dirichlet characters with a given large prime conductor $m$, where $P^{(a,b)}(z)=z^a{\bar{z}}^b$.