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 Fq[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 Fq, 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.

In 1991, Baker and Harman proved, under the assumption of the generalized Riemann hypothesis, thatThe purpose of this note is to deduce an analogous bound in the context of polynomials over a finite field using Weil's Riemann Hypothesis for curves over a finite field. Our approach is based on the work of Hayes who studied exponential sums over irreducible polynomials.

We prove a function field analogue of Maynard's celebrated result about primes with restricted digits. That is, for certain ranges of parameters the parameters n and q, we prove an asymptotic formula for the number of irreducible polynomials of degree n over a finite field Fq whose coefficients are restricted to lie in a given subset of Fq.

In 1991, Baker and Harman proved, under the assumption of the generalized Riemann hypothesis, that max θ∈[0,1) n x µ(n)e(nθ) ≪ǫ x 3/4+ǫ . The purpose of this note is to deduce an analogous bound in the context of polynomials over a finite field using Weil's Riemann Hypothesis for curves over a finite field. Our approach is based on the work of Hayes who studied exponential sums over irreducible polynomials.