Title: The function field Sathé-Selberg formula in arithmetic progressions and `short intervals',
Abstract: We use a function field analogue of a method of Selberg to derive an
asymptotic formula for the number of (square-free) monic polynomials in
$\mathbb{F}_q[X]$ of degree $n$ with precisely $k$ irreducible factors, in the
limit as $n$ tends to infinity. We then adapt this method to count such
polynomials in arithmetic progressions and short intervals, and by making use
of Weil's `Riemann hypothesis' for curves over $\mathbb{F}_q$, obtain better
ranges for these formulae than are currently known for their analogues in the
number field setting. Finally, we briefly discuss the regime in which $q$ tends
to infinity.