Sam Porritt

Chebyshev-type biases in function fields

Let Q be a fixed polynomial over a fixed finite field. Devin and Meng have recently established that polynomials f with large degree and with k irreducible divisors are in some precise sense biased towards quadratic residues mod Q if k is even and quadratic non residues if k is odd. We will explore this phenomenon as k varies with the degree of f. The main result is an explicit formula uniform in the variable k which shows that the bias disappears as k tends to infinity but not if it grows too quickly relative to the degree of f. The proof uses a modification of the Landau-Selberg-Delange method.