# A survey of the additive dilogarithm

## TL;DR

This paper surveys the additive dilogarithm and various versions of the weight two regulator in the infinitesimal setting, highlighting their arithmetic and geometric significance through a historical perspective.

## Contribution

It provides a comprehensive overview of the additive dilogarithm and different regulator constructions, connecting classical polylogarithms with infinitesimal approaches.

## Key findings

- Connections between additive dilogarithm and regulators clarified
- Historical approach enhances understanding of definitions and constructions
- Multiple versions of weight two regulator discussed

## Abstract

Borel's construction of the regulator gives an injective map from the algebraic $K$-groups of a number field to its Deligne-Beilinson cohomology groups. This has many interesting arithmetic and geometric consequences. The formula for the regulator is expressed in terms of the classical polyogarithm functions. In this paper, we give a survey of the additive dilogarithm and the several different versions of the weight two regulator in the infinitesimal setting. We follow a historical approach which we hope will provide motivation for the definitions and the constructions.