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

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 , ηχ_n) and that of L(¹₂ , χ_n) for χ_n ∈ O_n, where f is a monic irreducible polynomial in A = F_q[t], F_q is the finite field of characteristic p, χ_n : (A/fⁿA)^∗ → C^∗ is a character with some p-power order, O_n is the set of all the primitive cyclotomic characters χ_n 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.