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

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