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.