Matrices related to Dirichlet series

We attach a certain n×n matrix An to the Dirichlet series . We study the determinant, characteristic polynomial, eigenvalues, and eigenvectors of these matrices. The determinant of An can be understood as a weighted sum of the first n coefficients of the Dirichlet series L(s)−1. We give an interpretation of the partial sum of a Dirichlet series as a product of eigenvalues. In a special case, the determinant of An is the sum of the Möbius function. We disprove a conjecture of Barrett and Jarvis regarding the eigenvalues of An.