The probability that the number of points on a complete intersection is squarefree

Abstract

We consider the asymptotic probability that integers chosen according to a binomial distribution will have certain properties: (i) that such an integer is not divisible by the $k$th power of a prime, (ii) that any $k$ of $s$ chosen integers are relatively prime and (iii) that a chosen integer is prime. We also prove an analog of the Dirichlet divisor problem for the binomial distribution. We show how these results yield corresponding facts concerning the number of points on a smooth complete intersection over a finite field.