Ties in Function Field Prime Races
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908935907180544 |
|---|---|
| author | Bates, Graeme Jesubalan, Ryan Lee, Seewoo Lu, Jane Shim, Hyewon |
| author_facet | Bates, Graeme Jesubalan, Ryan Lee, Seewoo Lu, Jane Shim, Hyewon |
| contents | The function field analogue of Chebyshev's bias was first studied by Cha. In this paper, we study *ties* in this race, namely collections of distinct congruence classes $c_1, \dots, c_k \in (\mathbb{F}_q[T] / m)^\times$ for which $$π(N; m, c_1) = π(N; m, c_2) = \dots = π(N; m, c_k)$$ holds for infinitely many $N$. We provide infinitely many examples of $(m, c_1, \dots, c_k)$ for which the tie holds whenever $N$ satisfies certain congruence conditions. We give two different proofs: first, via the explicit formula for prime counts in terms of $L$-functions together with a matrix analogue of Möbius inversion, where exceptional pairs of Galois-conjugate elements in the corresponding cyclotomic fields produce ties; and second, via an explicit bijection arising from the $\mathrm{GL}_2(\mathbb{F}_q)$-action. Our examples also include characteristic 2 cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21005 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ties in Function Field Prime Races Bates, Graeme Jesubalan, Ryan Lee, Seewoo Lu, Jane Shim, Hyewon Number Theory 11T06, 11M38 The function field analogue of Chebyshev's bias was first studied by Cha. In this paper, we study *ties* in this race, namely collections of distinct congruence classes $c_1, \dots, c_k \in (\mathbb{F}_q[T] / m)^\times$ for which $$π(N; m, c_1) = π(N; m, c_2) = \dots = π(N; m, c_k)$$ holds for infinitely many $N$. We provide infinitely many examples of $(m, c_1, \dots, c_k)$ for which the tie holds whenever $N$ satisfies certain congruence conditions. We give two different proofs: first, via the explicit formula for prime counts in terms of $L$-functions together with a matrix analogue of Möbius inversion, where exceptional pairs of Galois-conjugate elements in the corresponding cyclotomic fields produce ties; and second, via an explicit bijection arising from the $\mathrm{GL}_2(\mathbb{F}_q)$-action. Our examples also include characteristic 2 cases. |
| title | Ties in Function Field Prime Races |
| topic | Number Theory 11T06, 11M38 |
| url | https://arxiv.org/abs/2603.21005 |