Convergence of families of Dirichlet series

TL;DR

This paper establishes conditions where uniform convergence of Dirichlet series ensures convergence of their coefficients and exponents, with applications to spectral zeta functions in geometry and physics.

Contribution

It provides new criteria linking uniform convergence of Dirichlet series to the convergence of their parameters, with practical applications in spectral theory.

Findings

Uniform convergence implies coefficient convergence under certain conditions

Applications to spectral zeta functions in geometry and physics

Conditions established for convergence of exponents and coefficients

Abstract

We give some conditions under which (uniform) convergence of a family of Dirichlet series to another Dirichlet series implies the convergence of their individual coefficients and/or exponents. We give some applications to some spectral zeta functions that arise in Riemannian geometry and physics.