Universality of L-functions over function fields

Abstract

We prove that Dirichlet L-functions corresponding to Dirichlet characters for $F_q\left[x\right]$ with q odd are universal in the following sense. Let $Q$ denote either the set of all prime polynomials Q in $F_q\left[x\right]$, or the set of all polynomials Q that are products of a fixed set of prime polynomials $Q_1,Q_2,\dots ,Q_r\in F_q\left[x\right]$. 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 $\frac{1}{2}<{\sigma }_1<{\sigma }_2<1$ and $0<\beta -\alpha \le 2\pi /(3\log q)$. 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 $Q\in Q$ 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.