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

TL;DR

This paper establishes explicit lower bounds for the absolute value of Dirichlet L-functions at 1, improving previous results through optimized trigonometric polynomial constructions.

Contribution

It provides new explicit lower bounds for |L(1, χ)| for large and all conductors q, using simulated annealing to optimize trigonometric polynomials.

Findings

Lower bound for large q: |L(1, χ)| ≥ 1/(9.12255 log(q/π))

Universal lower bound for q ≥ 2: |L(1, χ)| ≥ 1/(9.69030 log(q/π))

Improves previous bounds by Louboutin

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 $|L(1, \chi)| \geq 1/(9.12255 \log(q/\pi))$ for sufficiently large $q$, and that $|L(1, \chi)| \geq 1/(9.69030 \log(q/\pi))$ for all $q\geq2$, improving some results of Louboutin. The improvements stem principally from the construction, via simulated annealing, of some real trigonometric polynomials having particularly favorable properties.