A newcomer's guide to zeta functions of groups and rings

TL;DR

This paper provides an introductory overview of zeta functions associated with groups and rings, explaining their definitions, properties, and significance in understanding the structure and enumeration of these algebraic objects.

Contribution

It offers a comprehensive beginner-friendly guide to zeta functions in group and ring theory, synthesizing existing knowledge and clarifying complex concepts for new researchers.

Findings

Clarifies the definition and basic properties of zeta functions for groups and rings.

Highlights the role of zeta functions in counting substructures within algebraic objects.

Provides foundational knowledge for further research in asymptotic enumeration and algebraic zeta functions.

Abstract

These notes grew out of lectures given at the LMS-EPSRC Short Course on Asymptotic Methods in Infinite Group Theory, University of Oxford, 9-14 September 2007, organised by Dan Segal.