Statistical non-vanishing over function fields for arbitrary critical-line points

Establish that for every odd prime power \(q\), every point \(s\) on the critical line, and squarefree polynomials \(D\in\mathbb F_q[T]\) of degree \(n\), the proportion \(\delta_q(s;n)\) for which \(L(s,\chi_D)\neq 0\) converges to \(1\) as \(n\to\infty\).

Background

The authors formulate a function-field analogue of the expected 100% non-vanishing phenomenon. For an odd prime power $q$ and a point $s$ on the critical line, $\delta_q(s;n)$ is defined as the proportion of squarefree degree-$n$ polynomials $D$ whose associated quadratic Dirichlet $L$-function does not vanish at $s$. The conjecture predicts convergence to one as the genus or degree tends to infinity. The paper proves the central-point case for $q\equiv1\pmod 8$, leaving the stated general formulation unresolved.