Around a question of Baker

TL;DR

This paper reviews the history, current knowledge, and mathematical challenges surrounding Baker's question on the linear independence over rationals of special values of Dirichlet L-series at 1, focusing on non-trivial characters.

Contribution

It provides a comprehensive overview of Baker's question, including its origins, known results for prime moduli, and the obstacles for general moduli, along with potential generalizations.

Findings

Confirmed linear independence for prime moduli

Identified obstructions for arbitrary moduli

Summarized recent progress and open problems

Abstract

For any positive integer $q$, it is a question of Baker whether the numbers $L(1, \chi)$, where $\chi$ runs over the non-trivial characters mod $q$, are linearly independent over $\mathbb{Q}$. The question is answered in affirmative for primes but is unknown for an arbitrary modulus. In this expository note, we give an overview of the origin and history of this question as well as the state-of-the-art. We also give an account of the mathematical ideas that enter into this theme as well as elucidate the obstructions that preclude us from answering the question for arbitrary modulus. We also describe a number of generalizations and extensions of this question.