A function-field analogue of the Goldbach counting function and the associated Dirichlet series

TL;DR

This paper explores a function-field analogue of the Goldbach counting function's Dirichlet series, analyzing its meromorphic continuation and establishing the presence of a natural boundary in certain cases.

Contribution

It introduces a novel function-field analogue of the Goldbach counting function and investigates its associated Dirichlet series' analytic properties.

Findings

The Dirichlet series can sometimes be meromorphically continued to the entire plane.

In some cases, the series cannot be continued meromorphically, indicating a natural boundary.

The paper characterizes when the natural boundary occurs.

Abstract

We consider a function-field analogue of Dirichlet series associated with the Goldbach counting function, and prove that it can, or cannot, be continued meromorphically to the whole plane. When it cannot, we further prove the existence of the natural boundary of it.