Moments of Higher Logarithmic Derivatives and Exceptional Zeros in Cyclotomic Fields

Abstract

Let $\chi$ be a non-principal Dirichlet character, and let $L(s,\chi )$ be the associated Dirichlet $L$-function. We write $\mathcal{L}(s,\chi )$ for its logarithmic derivative $L^{\prime }(s,\chi )/L(s,\chi )$. In this article, we prove arithmetic formulas for the higher derivatives ${\mathcal{L}}^{(r)}(1,\chi )$ and establish unconditional moment asymptotics for $P^{(a,b)}({\mathcal{L}}^{(r)}(1,\chi ))$ as $\chi$ runs over non-principal Dirichlet characters of large prime conductor. As an application, we show that these moment asymptotics imply positivity of the second Li coefficient of prime cyclotomic fields of sufficiently large conductor. Combined with a zero-free criterion in terms of this Li coefficient, this rules out the possible exceptional zero in a Stark-type zero-free region for these fields.