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 $n$?



Converse to a theorem of Landau on Dirichlet series
I'm not sure you can hope for much. For example consider the case $c\_n=1$ if $n$ is not a square, and $c\_n=-1$ otherwise. The associated Dirichlet series has a pole at $s=1$, but of course the terms are not of fixed sign for sufficiently large $n$.


In general, knowing analytic properties of a Dirichlet series (such as convergence) cannot tell you much about any of the individual terms $c\_n$, since you can always change infinitely many of the $c\_n$ for $n$ in a "sparse" set.


