## Abstract

An old open problem in number theory is whether the Chebotarev density theorem holds in short intervals. More precisely, given a Galois extension E of Q with Galois group G, a conjugacy class C in G, and a 1 ≥ ϵ > 0, one wants to compute the asymptotic of the number of primes x ≤ p ≤ x+xϵ with Frobenius conjugacy class in E equal to C. The level of difficulty grows as ϵ becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime 1 ≥ ϵ > 1/2. We establish a function field analogue of the Chebotarev theorem in short intervals for any ϵ > 0. Our result is valid in the limit when the size of the finite field tends to ∞ and when the extension is tamely ramified at infinity. The methods are based on a higher dimensional explicit Chebotarev theorem and applied in a much more general setting of arithmetic functions, which we name G-factorization arithmetic functions.

Original language | English |
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Pages (from-to) | 597-628 |

Number of pages | 32 |

Journal | Transactions of the American Mathematical Society |

Volume | 373 |

Issue number | 1 |

DOIs | |

State | Published - 2020 |