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

Abstract

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.