Summary of Dirichlet Series Convergence from Apostol's IANT

Question

I've been trying to learn about Dirichlet series, in particular from Apostol's IANT textbook.

The textbooks tend to present result and not discuss them narratively, so I am left unsure of my correct understandanding of the results.

Are the following summary statements a correct interpretation of Apostol's IANT:

1. If a Dirichlet series $\sum_{n \leq x} a_n/n^s$ is bounded at complex $s_0=\sigma_0+it_0$ then it is also bounded at $s_1=\sigma_1+it_1$ where $\sigma_1>\sigma_0$.
2. If a Dirichlet series $\sum_{n \leq x} a_n/n^s$ is bounded at complex $s_0=\sigma_0+it_0$ then it also converges when $x\rightarrow \infty$ at $s_1=\sigma_1+it_1$ where $\sigma_1>\sigma_0$.
3. If $\sum_{n \leq x} a_n$ is bounded, then the Dirichlet series $\sum a_n/n^s$ converges for $\sigma>0$.