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

TL;DR

This paper generalizes Landau's theorem for Dirichlet series by relaxing the non-negativity condition on coefficients, allowing for controlled oscillations while still ensuring a singularity at the abscissa of convergence.

Contribution

It introduces a new framework that relaxes the non-negative coefficient condition using convex cones and polar cones, extending Landau's theorem to broader classes of Dirichlet series.

Findings

Established conditions under which Dirichlet series with oscillating coefficients have singularities

Quantified the freedom in choosing phase angles of coefficients

Proved sharpness of the generalized conditions

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.