Explicit lower bounds on |L(1,χ)|.

Let χ denote a primitive, non-quadratic Dirichlet character with conductor q , and let L (s , χ) denote its associated Dirichlet L -function. We show that | L (1 , χ) | ≥ 1 / (9.12255 log ⁡ (q / π)) for sufficiently large q , and that | L (1 , χ) | ≥ 1 / (9.69030 log ⁡ (q / π)) for all q ≥ 2 , improving some results of Louboutin. The improvements come from an averaging argument that simplifies Louboutin's approach, and the construction of certain trigonometric polynomials using simulated annealing, a technique often used in applied mathematics and computer science. We highlight the benefits of this approach to a pure mathematical problem. [ABSTRACT FROM AUTHOR]