Counting <i>w</i> -coprime <i>S</i> -integers and <i>S</i> -integral ideals in positive characteristic

Let [Formula: see text] be the finite field with [Formula: see text] elements, and let [Formula: see text] be an algebraic function field over [Formula: see text] whose field of constants is [Formula: see text]. Let [Formula: see text] be a finite nonempty set of prime divisors over [Formula: see text], and let [Formula: see text] be the ring of integers of [Formula: see text] attached to [Formula: see text]. Let [Formula: see text] be an integer. In this work we shall count [Formula: see text]-coprime [Formula: see text]-integers and [Formula: see text]-integral ideals, and our proofs are a combination of analytic methods, the Riemann-Roch theorem, and the Weil theorem for function fields in positive characteristic.