Singularities of ordinary Dirichlet series

Question

Is there an example of an ordinary Dirichlet series such that

(a) the Dirichlet series diverges to infinity at the real point (R > 0) on the line of convergence, and

(b) R is not a pole of the function represented by the Dirichlet series.

Answer 1

I think the following example answers my question: $$\sum_{n=1}^{\infty}[(-1)^{n+1}n^{-s+\frac{1}{2}}][\log \ n].$$