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Ties in function field prime race

Graeme Bates, Ryan Jesubalan, Seewoo Lee, Jane Lu, Hyewon Shim

Abstract

Function field analogue of the Chebyshev's bias was first studied by Cha. In this paper, we study ties in race, i.e. pairs of congruence classes $a, b \in (\mathbb{F}_q[T] / m)^\times$ where $π(N; m, a) = π(N; m, b)$ holds for infinitely many $N$. We provide infinitely many examples of $(m, a, b)$ where the tie holds for $N$ satisfying some certain congruence conditions. We give two different proofs, by 1) using the explicit formula via $L$-functions and matrix analogue of Möbius inversion formula, where exceptional Galois conjugate pairs of elements in the corresponding cyclotomic fields give ties, and 2) constructing an explicit bijection via $\mathrm{GL}_2(\mathbb{F}_q)$-action. Our examples also include characteristic 2 cases.

Ties in function field prime race

Abstract

Function field analogue of the Chebyshev's bias was first studied by Cha. In this paper, we study ties in race, i.e. pairs of congruence classes where holds for infinitely many . We provide infinitely many examples of where the tie holds for satisfying some certain congruence conditions. We give two different proofs, by 1) using the explicit formula via -functions and matrix analogue of Möbius inversion formula, where exceptional Galois conjugate pairs of elements in the corresponding cyclotomic fields give ties, and 2) constructing an explicit bijection via -action. Our examples also include characteristic 2 cases.
Paper Structure (31 sections, 9 theorems, 110 equations, 6 tables)

This paper contains 31 sections, 9 theorems, 110 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Arithmetic of function fields
  4. Chebyshev's bias
  5. Explicit counting formulas
  6. Matrix Möbius inversion and explicit counting formulas
  7. Examples
  8. m = T^2 + T + 1 over characteristic 2
  9. When $N$ is not a multiple of 3:
  10. When $N$ is a multiple of 3:
  11. m = T^3 + T + 1 over characteristic 2
  12. When $N$ is not a multiple of 7:
  13. When $N$ is a multiple of 7:
  14. m = T^2 + 1 over characteristic 3
  15. When $N$ is odd:
  16. ...and 16 more sections

Key Result

Proposition 3.1

The $(a, \chi)$-th entry of $\widetilde{Z}(n)$ is given by

Theorems & Definitions (19)

  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Remark 3.6
  • ...and 9 more