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arXiv · 2509.16142

Shanks' bias in function fields

Seewoo Lee
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Abstract

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

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Seewoo Lee. 2025-12-19. Shanks' bias in function fields. https://arxiv.org/abs/2509.16142

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