## Structural Properties of PolylogarithmsYears ago, the handful of peculiar numerical dilogarithmic identities, known since the time of Euler and Landen, gave rise to new discoveries concerning cyclotomic equations and related polylogarithmic ladders. These discoveries were made mostly by the methods of classical analysis, with help from machine computation. About the same time, starting with Bloch's studies on the application of the dilogarithm in algebraic $K$-theory and algebraic geometry, many important discoveries were made in diverse areas. This book seeks to provide a synthesis of these two streams of thought. In addition to an account of ladders and their association with functional equations, the chapters include applications to volume calculations in Lobatchevsky geometry, relations to partition theory, connections with Clausen's function, new functional equations, and applications to $K$-theory and other branches of abstract algebra. This rapidly-expanding field is brought up to date with two appendices, and the book concludes with an extensive bibliography of recent publications. About two-thirds of the material is accessible to mathematicians and scientists in many areas, while the remainder requires more specialized background in abstract algebra. |

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### Contents

The Evolution of the Ladder Concept | 1 |

Dilogarithmic Ladders | 11 |

Polylogarithmic Ladders | 27 |

Ladders in the TransKummer Region | 49 |

Supernumary Ladders | 69 |

Functional Equations and Ladders | 97 |

Multivariable Polylogarithm Identities | 123 |

Functional Equations of Hyperlogarithms | 171 |

tfTheory Cyclotomic Equations and Clausens Function | 233 |

Function Theory of Polylogarithms | 275 |

Partition Identities and the Dilogarithm | 287 |

The Dilogarithm and Volumes of Hyperbolic Polytopes | 301 |

Introduction to Higher Logarithms | 337 |

Some Miscellaneous Results | 355 |

Appendix A Special Values and Functional Equations | 377 |

Appendix B Summary of the Informal Polylogarithm Workshop | 401 |

### Common terms and phrases

5-units abelian algebraic functions algebraic integer algebraic number analytic arguments base equation Chapter Clausen Clausen's function coefficients complex component-ladders computation conjecture consider constant Corollary corresponding Coxeter cyclotomic equations cyclotomic equations satisfied Dedekind zeta functions defined definition degree denote differential dihedral dihedral angles dilogarithm dilogarithm function elements equivalent example expression factors field F finite formula functional equations given gives Hence higher logarithms Hodge structures hyperbolic hyperlogarithms identity independent integral inversion involving Kummer's equation Kummer's functional Lemma Lewin linear combinations linear power relations log(l log2 logarithmic terms MACSYMA mixed Hodge structures monodromy nontrivial number field Number Theory obtained orthoschemes polylogarithms polynomial polytopes prime Proof prove quadratic rational functions rational multiple resp Rogers root of unity Schlafli scissors congruence subgroup supernumary symmetry Theorem tions valid ladder values variables verified volume Zagier zero zeta functions