On an extension of a question of Baker

TL;DR

This paper extends known results on the linear independence over cb of special values of Dirichlet L-functions, covering arbitrary moduli and combining several previous independence results.

Contribution

It generalizes Baker, Birch, and Wirsing's linear independence results to arbitrary moduli and combines them with Okada and Murty-Murty's results, also proving independence for Erd
51sian functions with prime periods.

Findings

Sets of L(1, cb) values for certain moduli are linearly independent over cb.

Extended linear independence of cotangent values over cb.

Proved cb} linear independence for L-values of Erd
51sian functions with prime periods.

Abstract

It is an open question of Baker whether the numbers $L(1, \chi)$ for non-trivial Dirichlet characters $\chi$ with period $q$ are linearly independent over $\mathbb{Q}$. The best known result is due to Baker, Birch and Wirsing which affirms this when $q$ is co-prime to $\varphi(q)$. In this article, we extend their result to any arbitrary family of moduli. More precisely, for a positive integer $q$, let $X_q$ denote the set of all $L(1,\chi)$ values as $\chi$ varies over non-trivial Dirichlet characters with period $q$. Then for any finite set of pairwise co-prime natural numbers $q_i, 1\le i \le \ell$ with $(q_1 \cdots q_{\ell}, ~\varphi(q_1)\cdots \varphi(q_{\ell}))=1$, we show that the set $X_{q_1} \cup \cdots \cup X_{q_l}$ is linearly independent over $\mathbb{Q}$. In the process, we also extend a result of Okada about linear independence of the cotangent values over $\mathbb{Q}$ as well as a result of Murty-Murty about $\overline{\mathbb{Q}}$ linear independence of such $L(1, \chi)$ values. Finally, we prove $\mathbb{Q}$ linear independence of such $L$ values of Erd\"{o}sian functions with distinct prime periods $p_i$ for $1\le i \le \ell$ with $(p_1 \cdots p_{\ell}, ~ \varphi( p_1\cdots p_{\ell}) )= 1$.