On an extension of a question of Baker

Abstract

It is an open question of Baker whether the numbers [Formula: see text] for nontrivial Dirichlet characters [Formula: see text] with period [Formula: see text] are linearly independent over [Formula: see text]. The best known result is due to Baker, Birch and Wirsing which affirms this when [Formula: see text] is co-prime to [Formula: see text]. In this paper, we extend their result to any arbitrary family of moduli. More precisely, for a positive integer [Formula: see text], let [Formula: see text] denote the set of all [Formula: see text] values as [Formula: see text] varies over nontrivial Dirichlet characters with period [Formula: see text]. Then for any finite set of pairwise co-prime natural numbers [Formula: see text] with [Formula: see text] [Formula: see text], we show that the set [Formula: see text] is linearly independent over [Formula: see text]. In the process, we also extend a result of Okada about linear independence of the cotangent values over [Formula: see text] as well as a result of Murty–Murty about [Formula: see text] linear independence of such [Formula: see text] values. Finally, we prove [Formula: see text] linear independence of such [Formula: see text] values of Erdösian functions with distinct prime periods [Formula: see text] for [Formula: see text] with [Formula: see text] [Formula: see text].