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

Abstract

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 $L'(s, \chi)/L(s, \chi)$. 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 $P^{(a,b)}(z) = z^a \overline{z}^b$.