Does convergence/divergence in the real domain completely determine it in the complex domain for Dirichlet series?

Question

Is it sufficient to consider divergence only in the real domain to understand divergence of Dirichlet series in the complex domain?

If we can show that a Dirichlet series with complex parameter $s=\sigma+it$ converges for all $\sigma>\sigma_a$, then $\sigma_a$ completely determines the divergence and convergence behaviour of the series over all the complex domain.

Is this correct?

If yes, is this also correct for absolute and conditional convergence?

Answer 1

Yes and yes. The constant $\sigma_a$ is called abscissa of convergence and is defined as $\sigma_a = \displaystyle \inf \left\{ {\mathfrak{R} \left({s}\right) \colon s \in \mathbb C, \lvert F(s)\rvert<\infty }\right\},$ where $F$ is a dirichlet series. From this plain definition it is clear that we only care about the fact that is is convergent (in the whole complex plane), and not how it converges.