Ratios of Artin L-functions

TL;DR

This paper proves that specific ratios of Artin L-functions possess infinitely many poles, using a converse theorem for Maass forms that does not assume automorphy, relying only on known properties of these L-functions.

Contribution

It introduces a new approach to analyze ratios of Artin L-functions without assuming their automorphy, expanding understanding of their pole structure.

Findings

Certain quotients of Artin L-functions have infinitely many poles.

The proof uses a converse theorem for Maass forms with relaxed assumptions.

The result does not depend on the automorphy conjecture for Artin L-functions.

Abstract

We show that certain quotients of Artin L-functions have infinitely many poles. Our result follows from a converse theorem for Maass forms of Laplace eigenvalue 1/4 in which the twisted L-functions are not assumed to be entire. We do not require the conjectural automorphy of Artin L-functions, only their established meromorphic continuation and functional equation.