Universality of L-functions over function fields

We prove that Dirichlet L-functions corresponding to Dirichlet characters for F-q[x] with q odd are universal in the following sense. Let Q denote either the set of all prime polynomials Q in F-q[x], or the set of all polynomials Q that are products of a fixed set of prime polynomials Q(1), Q(2),..., Q(r) is an element of F-q[x]. Let U be the open rectangle with vertices sigma(1)+ i alpha, sigma(2)+i alpha, sigma(2)+i beta, sigma(1)+i beta, where 1/2 < sigma(1) < sigma(2) < 1 and 0 < beta - alpha <= 2 pi/(3 log q). Suppose also that C is a compact set in U with positive Lebesgue measure whose complement is connected and that f is a prescribed continuous, nonvanishing function on C that is analytic on the interior of C. Then if Q is an element of Q is of high enough degree, a positive proportion of the L-functions with characters to this modulus approximate f arbitrarily closely. This extends for the first time (as far as we know) the notion of universality of L-functions over number fields to the function field setting. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.