Matrices related to Dirichlet series

Abstract

We attach a certain $n\times n$ matrix $A_n$ to the Dirichlet series $L(s)={\sum }_{k=1}^{\infty }{a}_k{k}^{-s}$. We study the determinant, characteristic polynomial, eigenvalues, and eigenvectors of these matrices. The determinant of $A_n$ 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 $A_n$ is the sum of the Möbius function. We disprove a conjecture of Barrett and Jarvis regarding the eigenvalues of $A_n$.

17 pages