Chebotarev density theorem in short intervals for extensions of $\mathbb{F}_q(T)$

TL;DR

This paper proves a function field analogue of the Chebotarev density theorem in short intervals for any positive epsilon, extending previous results limited to larger epsilon values under GRH.

Contribution

It establishes the Chebotarev theorem in short intervals over function fields for all epsilon>0 without GRH, using higher dimensional explicit formulas and general G-factorization functions.

Findings

Valid in the limit as the finite field size tends to infinity

Applicable to tamely ramified extensions at infinity

Extends Chebotarev density results to all epsilon>0 in function fields

Abstract

An old open problem in number theory is whether Chebotarev density theorem holds in short intervals. More precisely, given a Galois extension $E$ of $\mathbb{Q}$ with Galois group $G$, a conjugacy class $C$ in $G$ and an $1\geq \varepsilon>0$, one wants to compute the asymptotic of the number of primes $x\leq p\leq x+x^{\varepsilon}$ with Frobenius conjugacy class in $E$ equal to $C$. The level of difficulty grows as $\varepsilon$ becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime $1\geq\varepsilon>1/2$. We establish a function field analogue of Chebotarev theorem in short intervals for any $\varepsilon>0$. Our result is valid in the limit when the size of the finite field tends to $\infty$ 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.