Estimates for Coefficients ofL-Functions for Function Fields

We consider the Dirichlet characters for polynomial rings q[T] and the associatedL-functions. By Weilʹs result, the associatedL-functions are all polynomials. Applying Burgessʹ idea, we obtain an upper bound for the coefficients of theseL-functions. As an application, using our estimates, we obtain an upper bound for the degree of the fundamental unit in real quadratic function fields.