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arXiv · 1709.01963

The function field Sath\'e-Selberg formula in arithmetic progressions and `short intervals'

Ardavan Afshar·Sam Porritt
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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.

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Ardavan Afshar, Sam Porritt. 2017-09-06. The function field Sath\'e-Selberg formula in arithmetic progressions and `short intervals'. https://doi.org/10.4064/aa170726-24-4

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