Shanks' bias in function fields

Abstract

We study the function field analogue of Shanks bias. For Liouville function $\lambda (f)$, we compare the number of monic polynomials $f$ with $\lambda (f){\chi }_m(f)=1$ and $\lambda (f){\chi }_m(f)=-1$ for a nontrivial quadratic character ${\chi }_m$ modulo a monic square-free polynomial $m$ over a finite field. Under Grand Simplicity Hypothesis (GSH) for $L$-functions, we prove that $\lambda \cdot {\chi }_m$ is biased towards $+1$. We also give some examples where GSH is violated.

Accepted version, to appear in Journal de Théorie des Nombres de Bordeaux