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

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 >= epsilon > 0, one wants to compute the asymptotic of the number of primes x <= p <= 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 >= epsilon > 1/2. We establish a function field analogue of the Chebotarev theorem in short intervals for any E > 0. Our result is valid in the limit when the size of the finite field tends to Do 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.