On an extension of a question of Baker (2212.00376v1)

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 $(p_1 \cdots p_{\ell}, ~ \varphi( p_1\cdots p_{\ell}) )= 1$.