If I want a power series $\sum_n a_n \, z^n$ that cannot be analytically continued anywhere beyond its disk of convergence $|z| < R$, then I can use a lacunary series, e.g., $\sum_n z^{2^n}$.
Are there any similar techniques for constructing a Dirichlet series $\sum_n a_n / n^s$ that cannot be analytically continued beyond its abscissa of absolute convergence?
(I'm sorry if this question is a bit open-ended, but I want to avoid getting a full answer to my homework question.)