Large values of Dirichlet L-functions over function fields

Abstract

In this paper, we investigate the existence of large values of |L(s,χ)|, where χ varies over non-principal characters associated to prime polynomials Q over finite field q, as d(Q) →∞, and s (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 |L(1/2,χ)|, where now χ varies over even, non-principal, Dirichlet characters associated to prime polynomials Q over Fq, as d(Q) →∞. In addition to resonance method, in this case, we use an adaptation of Gál-type sums estimate.