UNIFORM ESTIMATES FOR ALMOST PRIMES OVER FINITE FIELDS

We establish a new asymptotic formula for the number of polynomials of degree n with k prime factors over a finite field Fq. The error term tends to 0 uniformly in n and in q. Previously, asymptotic formulas were known either for fixed q, through the works of Warlimont [Arch. Math. (Basel) 60 (1993), pp. 58-72] and Hwang [Random Structures Algorithms 13 (1998), pp. 17-47], or for small k, through the work of Arratia, Barbour and Tavaré [Math. Proc. Cambridge Philos. Soc. 114 (1993), pp. 347-368]. As an application, we estimate the total variation distance between the number of cycles in a random permutation on n elements and the number of prime factors of a random polynomial of degree n over Fq. The distance tends to 0 at rate 1/(q^√log n). Previously this was only understood when either q is fixed and n tends to ∞, or n is fixed and q tends to ∞, by results of Arratia, Barbour and Tavaré.