Counting w-coprime S-integers and S-integral ideals in positive characteristic

Let Fq be the finite field with q elements, and K an algebraic function field over with Fq as its field of constants.

Let S be a finite nonempty set of prime divisors over K, and OS be the ring of integers of K attached to S.

Let w greater than 1 be an integer.

In this work we shall count w coprime S integers and S integral ideals, and our proofs are a combination of analytic methods and the Riemann Roch theorem and the Weil theorem for function fields in positive characteristic.