Matrices related to Dirichlet series

Details Matrices related to Dirichlet series Matrices related to Dirichlet series

2008-08-30

arxiv.org/abs/0809.0076v1

Mathematics

Number Theory

17 pages

Scientific paper

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 M\"obius function. We disprove a conjecture of Barrett and Jarvis regarding the eigenvalues of $A_n$.

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