Shanks' bias in function fields
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912777329704960 |
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| author | Lee, Seewoo |
| author_facet | Lee, Seewoo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16142 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shanks' bias in function fields Lee, Seewoo Number Theory 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. |
| title | Shanks' bias in function fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.16142 |