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

Number Theory