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Shanks' bias in function fields

Seewoo Lee

TL;DR

The paper analyzes the function-field analogue of Shanks’ bias for the Liouville function twisted by a nontrivial quadratic character modulo a monic square-free m over F_q[T]. Under the Grand Simplicity Hypothesis for L-functions, it proves a bias toward +1 for λ·χ_m and derives explicit density formulas for both non-cumulative and cumulative counts, linking the bias magnitude to central L-values. It also establishes that, under GSH, Möbius-based biases vanish for square-free m, and it provides concrete examples across deg m = 1,2,3 and a higher-degree case to illustrate the phenomena and potential failures when GSH is violated. The methodology combines rational L-functions, linear recurrences, and Kronecker–Weyl equidistribution to translate oscillatory bias terms into explicit densities. The results illuminate how function-field analogues of classical biases behave under symmetry hypotheses and offer computational illustrations with SageMath code.

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.

Shanks' bias in function fields

TL;DR

The paper analyzes the function-field analogue of Shanks’ bias for the Liouville function twisted by a nontrivial quadratic character modulo a monic square-free m over F_q[T]. Under the Grand Simplicity Hypothesis for L-functions, it proves a bias toward +1 for λ·χ_m and derives explicit density formulas for both non-cumulative and cumulative counts, linking the bias magnitude to central L-values. It also establishes that, under GSH, Möbius-based biases vanish for square-free m, and it provides concrete examples across deg m = 1,2,3 and a higher-degree case to illustrate the phenomena and potential failures when GSH is violated. The methodology combines rational L-functions, linear recurrences, and Kronecker–Weyl equidistribution to translate oscillatory bias terms into explicit densities. The results illuminate how function-field analogues of classical biases behave under symmetry hypotheses and offer computational illustrations with SageMath code.

Abstract

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

Paper Structure

This paper contains 12 sections, 14 theorems, 114 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chebyshev's bias and its variants
  4. Function fields, L-functions, and Chebyshev's bias
  5. Kronecker--Weyl Equidistribution theorem
  6. L-functions
  7. Density and biases
  8. Examples
  9. When deg m = 1
  10. When deg m = 2
  11. When deg m = 3
  12. Higher degree example

Key Result

Theorem 1

Let $A_{\pm}^\lambda(n;m)$ be the number of non-constant monic polynomials $f \in A$ of degree at most $n$ with $\lambda(f) \chi_m(f) = \pm 1$. Assume that GSH holds for the Dirichlet $L$-function $L(s, \chi_m)$. Then we have where a closed formula for the density is obtained. When $q \equiv 1 \pmod{4}$, the density can be made arbitrarily close to $\frac{1}{2}$ by choosing $m$ appropriately with

Theorems & Definitions (24)

  • Theorem : Corollary \ref{['cor:mainlambda']} and \ref{['cor:gapLfunc']}
  • Theorem 2.1: cha2008chebyshev
  • Theorem 2.2: Kronecker--Weyl
  • Corollary 2.3
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • ...and 14 more