  A classical theorem of Landau states that, if an ordinary Dirichlet series
has non-negative coefficients, then it has a singularity on the real line at
its abscissae of absolute convergence. In this article, we relax the condition
on the coefficients while still arriving at the same conclusion. Specifically,
we write $a_n$ as $|a_n| e^{i \ttt_n}$ and we consider the sequences $\{\;
|a_n| \; \}$ and $\{\; \cos{\ttt_n} \; \}$. Let $M \in \mathbb{N}$ be given.
The condition on $\{\; |a_n| \; \}$ is that, dividing the sequence sequentially
into vectors of length $M$, each vector lies in a certain convex cone $B
\subset [0,\infty)^M$. The condition on $\{\; \cos{\ttt_n} \; \}$ is (roughly)
that, again dividing the sequence sequentially into vectors of length $M$, each
vector lies in the negative of the polar cone of $B$. We attempt to quantify
the additional freedom allowed in choosing the $\ttt_n$, compared to Landau's
theorem. We also obtain sharpness results.
