# A note on abscissas of Dirichlet series

## TL;DR

This paper introduces an abstract framework for understanding the convergence properties of vector-valued Dirichlet series, revealing that certain abscissas are equal for Hardy spaces and exploring weak variants.

## Contribution

It provides a novel abstract approach to Dirichlet series abscissas and establishes equality of abscissas for Hardy spaces, along with new weak abscissa concepts.

## Key findings

- Abscissas for Hardy spaces of Dirichlet series are all equal.
- Introduces and studies weak versions of abscissas for scalar-valued Dirichlet series.
- Provides an abstract approach to convergence of vector-valued Dirichlet series.

## Abstract

We present an abstract approach to the abscissas of convergence of vector-valued Dirichlet series. As a consequence we deduce that the abscissas for Hardy spaces of Dirichlet series are all equal. We also introduce and study weak versions of the abscissas for scalar-valued Dirichlet series.