Chebotarev density theorem in short intervals for extensions of 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 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+x^ε 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.