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

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 $1\ge \epsilon >0$, one wants to compute the asymptotic of the number of primes $x\le p\le x+x^{\epsilon }$ with Frobenius conjugacy class in $E$ equal to $C$. The level of difficulty grows as $\epsilon$ becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime $1\ge \epsilon >1/2$. We establish a function field analogue of Chebotarev theorem in short intervals for any $\epsilon >0$. 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.

Incorporated referee comments. Accepted for publication in Trans. Amer. Math. Soc

References in corpus (4)

• Arithmetic functions in short intervals and the symmetric group
• An effective Chebotarev density theorem under GRH
• The variance of the number of prime polynomials in short intervals and in residue classes
• A polynomial analogue of Landau's theorem and related problems

Cited by in corpus (1)

• A geometric approach to counting norms in cyclic extensions of function fields