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 E of Q \mathbb {Q} with Galois group G G , a conjugacy class C C in G G , and a 1 ≥ ε > 0 1\geq \varepsilon >0 , one wants to compute the asymptotic of the number of primes x ≤ p ≤ x + x ε x\leq p\leq x+x^{\varepsilon } with Frobenius conjugacy class in E E equal to C C . The level of difficulty grows as ε \varepsilon becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime 1 ≥ ε > 1 / 2 1\geq \varepsilon >1/2 . We establish a function field analogue of the Chebotarev theorem in short intervals for any ε > 0 \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 G -factorization arithmetic functions.