NON-VANISHING of L-FUNCTIONS for CYCLOTOMIC CHARACTERS in FUNCTION FIELDS

Abstract

In the number field case, it is conjectured that the central values L(12 , χ) of L-functions are nonzero, where χ : (Z/mZ)∗ → C∗ is a primitive Dirichlet character with conductor m. We resolve this conjecture in the function field case by proving that there are infinitely many cyclotomic characters for which the central values of L-functions are nonzero. In detail, for a given positive integer n, we compute the mean value of L(12 , ηχn) and that of L(12 , χn) for χn ∈ On, where f is a monic irreducible polynomial in A = Fq[t], Fq is the finite field of characteristic p, χn : (A/fnA)∗ → C∗ is a character with some p-power order, On is the set of all the primitive cyclotomic characters χn modulo fn with p-power order, g is a monic polynomial in A that is relatively prime to f, and η : (A/gA)∗ → C∗ is a primitive even character. 2021 American Mathematical Society