Matrices related to Dirichlet series
by
David Cardon
Brigham Young University

We attach a certain matrix n ×n matrix A_n to the Dirichlet series L(s)=∑_k=1^∞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 twisted sum of the first n terms 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.

Date received: June 16, 2008