Abscissa of absolute convergence of a Dirichlet series (${\sigma }_a=\inf \{\sigma :{\sum }_n|a_n|n^{-\sigma }<\infty \}$)

absolute convergence of a Dirichlet series depends only on the real part of its variable. The set of real parts giving convergence is upward closed by the comparison test. If that set is nonempty and not all of the real line, its finite infimum is the abscissa: every larger real part gives convergence, and every smaller real part gives divergence. Boundary convergence is a separate question.