Matrices related to Dirichlet series

TL;DR

This paper introduces a family of matrices associated with Dirichlet series, analyzing their algebraic properties and providing new interpretations of sums and eigenvalues related to these series.

Contribution

It defines matrices linked to Dirichlet series and explores their determinants, eigenvalues, and eigenvectors, including disproving a prior conjecture about their eigenvalues.

Findings

Determinant of $A_n$ relates to coefficients of the inverse Dirichlet series.

Partial sums of Dirichlet series interpreted as products of eigenvalues.

Disproved a conjecture on eigenvalues of $A_n$.

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