Counting Subrings of ${\mathbb Z}^n$ of finite index

TL;DR

This paper studies the enumeration of subrings within the integer lattice ^d by employing subring zeta functions and p-adic integration techniques to understand their distribution and count.

Contribution

It introduces a novel approach combining zeta functions and p-adic methods to count subrings of ^d, advancing the understanding of their structure and enumeration.

Findings

Derived formulas for counting subrings of ^d

Connected subring counts to properties of zeta functions

Provided new insights into the distribution of subrings

Abstract

In this article we investigate the number of subrings of $\Z^d$ using subring zeta functions and $p$-adic integration.