Statistics of Prime Divisors in Function Fields

ROBERT C. RHOADES Abstra t. We show that the prime divisors of a random polynomial in Fq[t] are typi ally Poisson Distributed . This result is analogous to the result in Z of Granville [1℄. Along the way, we use a sieve developed by Granville and Soundararajan [2℄ to give a simple proof of the Erdös-Ka theorem in the fun tion eld setting. This approa h gives stronger results about the moments of the sequen e {ω(f)}f∈Fq [t] than was previously known, where ω(f) is the number of prime divisors of f . 1. Introdu tion Let ω(n) denote the number of distin t prime divisors of an integer n. A elebrated theorem of Erdös and Ka is that for α ≤ β one has