The function field Sathe–Selberg formula in arithmetic progressions and `short intervals'
Volume 187 / 2019
Acta Arithmetica 187 (2019), 101-124
MSC: Primary 11T55; Secondary 11M38.
DOI: 10.4064/aa170726-24-4
Published online: 2 January 2019
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 $\mathbb{F}_q$, we 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.
