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

TL;DR

This paper develops methods to count w-coprime S-integers and S-integral ideals in algebraic function fields over finite fields, combining analytic techniques with classical theorems in positive characteristic.

Contribution

It introduces a novel approach that merges analytic methods with algebraic geometry to count specific algebraic objects in positive characteristic function fields.

Findings

Derived formulas for counting w-coprime S-integers

Established bounds for S-integral ideals

Combined analytic and geometric techniques for counting

Abstract

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.