Large values of Dirichlet L-functions over function fields

Abstract

In this paper, we investigate the existence of large values of [Formula: see text], where [Formula: see text] varies over non-principal characters associated to prime polynomials [Formula: see text] over finite field [Formula: see text], as [Formula: see text], and [Formula: see text]. When [Formula: see text], we provide a lower bound for the number of such characters. To do this, we adapt the resonance method to the function field setting. We also investigate this problem for [Formula: see text], where now [Formula: see text] varies over even, non-principal, Dirichlet characters associated to prime polynomials [Formula: see text] over [Formula: see text], as [Formula: see text]. In addition to resonance method, in this case, we use an adaptation of Gál-type sums estimate.