Large values of Dirichlet L-functions over function fields

In this paper, we investigate the existence of large values of vertical bar L(s, chi)vertical bar, where. varies over non-principal characters associated to prime polynomials Q over finite field F-q, as d(Q) -> infinity, and s is an element of(1/2, 1]. When s = 1, 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 vertical bar L(1/2, chi)vertical bar, where now. varies over even, non-principal, Dirichlet characters associated to prime polynomials Q over F-q, as d(Q) -> infinity. In addition to resonance method, in this case, we use an adaptation of Gal-type sums estimate.