A formula of counting divisors in integers rings: a generalization of the divisor function $d_0(n)$

TL;DR

This paper generalizes the classical divisor function to Dedekind domains with finite class groups by establishing a formula for counting divisors via zero-sum subsequences and ideal decompositions.

Contribution

It introduces a closed-form formula using character theory to count divisors in integer rings with cyclic class groups, extending the divisor function concept.

Findings

Derived a structural theorem for ideals in cyclic class group rings.

Established a formula for counting zero-sum subsequences in finite abelian groups.

Characterized irreducible elements based on zero-sum subsequence counts.

Abstract

In this paper we establish a formal connection between the structure of ideals in integers rings and the theory of additive combinatorics. For integers rings with cyclic class groups, we prove a structural theorem demonstrating that every non-zero ideal can be decomposed into a maximal principal part and a product of ideals whose total length is bounded by the Davenport constant. With this decomposition we find divisors for generators of the ideal $I=(\alpha, \beta)$. The central result of this work is the derivation of a closed formula using character theory over finite abelian groups to count the exact number of zero-sum subsequences of a given sequence. Under the established correspondence between principal ideals and zero-sum sequences, this formula provides a precise counting of the principal ideal divisors of any given ideal, and therefore counting common divisors of generators of the ideal $I=(\alpha,\beta)$. This result constitutes a natural generalization of the classical divisor function $d_0(n)$ from unique factorization domains to any Dedekind domain with a finite class group. Finally, we characterize irreducible elements in $\mathcal{O}_K$ based on the counting of these zero-sum subsequences.