Chowla’s theorem over function fields

Abstract

Sarvadaman Chowla proved that if [Formula: see text] is an odd prime, then [Formula: see text] ([Formula: see text]) are linearly independent over the field of rational numbers. We establish an analog of this result over function fields. As an application, we prove an analog of the Baker–Birch–Wirsing theorem about the non-vanishing of Dirichlet series with periodic coefficients at [Formula: see text] in the function field setup with a parity condition.