  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 $1\geq
\varepsilon>0$, 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 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.
