Extending Landau's Theorem on Dirichlet Series with Non-Negative Coefficients

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 abscissa of convergence. In this article, we relax the condition on the coefficients while still arriving at the same conclusion. Specifically, we write $a_n$ as $|a_n| e^{i \theta _n}$ and we consider the sequences $\{ |a_n| \}$ and $\{ \cos{\theta _n} \}$. 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 $\{ \cos{\theta _n} \}$ 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 demonstrate the additional freedom allowed in choosing the $\theta _n$, compared to Landau's Theorem. We also obtain sharpness results.