  We prove an analogue of the Prime Number Theorem for short intervals on a
smooth projective geometrically irreducible curve of arbitrary genus over a
finite field. A short interval "of size E" in this setting is any additive
translate of the space of global sections of a sufficiently positive divisor E
by a suitable rational function f. Our main theorem gives an asymptotic count
of irreducible elements in short intervals on a curve in the "large q" limit,
uniformly in f and E. This result provides a function field analogue of an
unresolved short interval conjecture over number fields, and extends a theorem
of Bary-Soroker, Rosenzweig, and the first author, which can be understood as
an instance of our result for the special case of a divisor E supported at a
single rational point on the projective line.
