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

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 $\mathbb {Q}$ with Galois group $G$, a conjugacy class $C$ in $G$, and a $1\geq \varepsilon >0$, 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 the 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.