Landau's Theorem for Dirichlet series with real coefficients ($c_n$) states that if the coefficients are of fixed sign for all sufficiently large $n$, then the point $\sigma_0$ on the abscissa of convergence of the series is itself a singularity of the function represented by the series in the half plane $\sigma>\sigma_0$.

One can also show that the conclusion of Landau's Theorem applies in a broader context. For example, if the partial sums $\sum_{n\leq x} c_n\geq 0$, then the real point $\sigma_0$ is a singularity of the function.

My question is this: To what extent does the converse implication hold, that is, if $\sigma_0$ is a singularity of the function represented for $\sigma>\sigma_0$ by some Dirichlet series with real coefficients, then under what additional conditions may we conclude that the ($c_n$) are of fixed sign for sufficiently large $n$?