TITLE: Divergence of Dirichlet series
QUESTION [7 upvotes]: Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge?

REPLY [3 votes]: I have worked out a partial answer to my own question.
Define $A(s) = \sum\limits_{n=1}^\infty\frac{a_n}{n^s}$.
First, assume that $s = 1$.  WLOG we can also assume that $a = 1$.  Observe that for some $N$ and all $n > N$, $\Re(a_n) > 1/2$.  It follows that $A(1)$ diverges.
I found a pertinent result in Knopp, Konrad. Infinite Sequences and Series. New York: Dover, 1956. Item 4 on p. 138 is the result that if the partial sums of the Dirichlet series $A(s_0)$ are bounded, then $A(s)$ converges provided $\Re(s) > \Re(s_0)$.  Since $A(1)$ diverges, we conclude from this result that $A(s_0)$ diverges for $s_0 < 1$.
Thus, we see that $A(s)$ must diverge for $\Re(s) \in (0,1]$, unless $s = 1 + it$, where $t \neq 0$.
Please see my question on MO for a demonstration of divergence in the remaining case.