Explicit lower bounds on $|L(1,\chi )|$

Abstract

Let $\chi$ denote a primitive, non-quadratic Dirichlet character with conductor $q$, and let $L(s,\chi )$ denote its associated Dirichlet $L$-function. We show that $|L(1,\chi )|\ge 1/(9.12255\log (q/\pi ))$ for sufficiently large $q$, and that $|L(1,\chi )|\ge 1/(9.69030\log (q/\pi ))$ for all $q\ge 2$, improving some results of Louboutin. The improvements stem principally from the construction, via simulated annealing, of some real trigonometric polynomials having particularly favorable properties.

13 pages