Prime specialization in higher genus I

Abstract

A classical conjecture predicts how often a polynomial in takes prime values. The natural analogous conjecture for prime values of a polynomial f(T) ∈ k[u][T], where k is a finite field, is false. The conjecture over k[u] was modified in earlier work by introducing a correction factor that encodes unexpected periodicity of the Möbius function at the values of f on k[u] when f ∈ k[u][Tp], where p is the characteristic of k.