NON-VANISHING OF L-FUNCTIONS FOR CYCLOTOMIC CHARACTERS IN FUNCTION FIELDS

초록

In the number field case, it is conjectured that the central values L(1/2, chi) of L-functions are nonzero, where chi : (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(1/2, eta chi(n)) and that of L(1/2, chi(n)) for chi(n) is an element of O-n, where f is a monic irreducible polynomial in A = F-q[t], F-q is the finite field of characteristic p, chi(n) : (A/f(n) A)* -> C* is a character with some p-power order, O-n is the set of all the primitive cyclotomic characters chi(n) modulo f(n) with p-power order, g is a monic polynomial in A that is relatively prime to f, and eta : (A/gA)* -> C* is a primitive even character.

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