Polylogarithms, Harmonic Polylogarithms, and Multiple Zeta Values

Exploratory code and notebooks studying the hierarchy of polylogarithmic special functions used in

• number theory
• algebraic geometry
• perturbative quantum field theory
• multiloop Feynman integral computations

The project focuses on the relationships between

• logarithms
• classical polylogarithms Liₙ(x).
• harmonic polylogarithms (HPLs)
• multiple polylogarithms (MPLs / generalized polylogarithms)
• multiple zeta values (MZVs)

and the algebraic structures connecting them:

• shuffle algebras
• stuffle (quasi-shuffle) algebras
• double-shuffle relations
• Hopf algebra structures and coproducts

The repository combines mathematical exposition in Jupyter notebooks with Python experimentation.

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Repository structure

harmonic-polylog
│
├── pyproject.toml        Project configuration (uv / PEP 621)
├── hpl.py                Experimental HPL utilities
│
├── theory.ipynb          Mathematical theory and derivations
├── exploration.ipynb     Numerical exploration and experiments
│
└── README.md

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Mathematical hierarchy of functions

These functions arise from the theory of iterated integrals.

flowchart TD

A["Iterated integrals"]

A --> B["Multiple polylogarithms (MPLs) / generalized polylogarithms"]

B --> C["Harmonic polylogarithms (HPLs)"]

C --> D["Classical polylogarithms Li_n(x)"]

D --> E["Logarithms"]

B --> F["Special values"]
F --> G["Multiple zeta values (MZVs)"]

B --> H["Single-valued polylogarithms"]

A --> I["Beyond polylogarithms"]

I --> J["Elliptic polylogarithms"]

J --> K["Higher-genus iterated integrals"]

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Interpretation:

• MPLs form the central class of polylogarithmic functions.
• HPLs are a restricted alphabet widely used in physics.
• MZVs arise as special values of MPLs.
• More complicated Feynman integrals require elliptic polylogarithms.

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Algebraic structures

Multiple zeta values possess two compatible product structures.

flowchart LR

A["Multiple zeta values"]

A --> B["Iterated integral representation"]
A --> C["Nested series representation"]

B --> D["Shuffle algebra"]

C --> E["Stuffle algebra"]

D --> F["Double-shuffle relations"]
E --> F

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Explanation:

• Shuffle relations arise from iterated integrals.
• Stuffle relations arise from nested sums.
• Double-shuffle relations arise from requiring both descriptions to agree.

These relations generate many identities among MZVs.

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Notebooks

theory.ipynb

A conceptual overview covering

• iterated integrals
• classical polylogarithms
• harmonic polylogarithms
• multiple polylogarithms
• multiple zeta values
• shuffle and stuffle algebras
• double-shuffle relations
• Hopf algebra structure
• polylogarithms in Feynman integrals

Includes a worked example: ζ(2) ζ(3) = ζ(2,3) + ζ(3,2) + ζ(5) expanded using shuffle and stuffle products.

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exploration.ipynb

Interactive experiments including

• numerical evaluation of polylogarithms
• exploring identities among MZVs
• symbolic checks using SymPy
• plotting special functions

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Python module

hpl.py contains experimental utilities for working with harmonic polylogarithms.

Features include

• recursive HPL definitions
• symbolic manipulation
• numerical evaluation using mpmath

This module is currently intended for experimentation, not production use.

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Installation

This project uses uv for Python environment management.

Install uv

Mac (Homebrew)

brew install uv

or

pip install uv

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Create environment

From the repository root:

uv venv
uv pip install -e .

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Launch notebooks

uv run jupyter lab

or

uv run jupyter notebook

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Example usage

Evaluate a classical polylogarithm:

import mpmath as mp

mp.polylog(2, 0.5)

Evaluate a zeta value:

mp.zeta(3)

which corresponds to the multiple zeta value ζ(3), known as Apéry’s constant.

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Why these functions matter

In modern perturbative quantum field theory, many multiloop integrals evaluate to generalized polylogarithms.

Typical workflow:

1. derive differential equations for master integrals
2. transform them into canonical form
3. integrate as iterated integrals

which naturally yields multiple polylogarithms.

More complicated integrals require extensions to

• elliptic polylogarithms
• modular iterated integrals
• higher-genus geometries.

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References

Chen (1977) — Iterated path integrals. Bulletin AMS

Remiddi & Vermaseren (2000) — Harmonic polylogarithms. Int. J. Mod. Phys. A

Zagier (1994) — Values of zeta functions and their applications.

Goncharov (2001) — Multiple polylogarithms and mixed Tate motives.

Duhr (2012) — Hopf algebras, coproducts and generalized polylogarithms. JHEP

Brown (2012) — Mixed Tate motives over ℤ. Annals of Mathematics.

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Future directions

Possible extensions for this project:

• implementing a generalized polylogarithm evaluator
• computing higher-depth MZVs
• implementing explicit shuffle and stuffle algebra
• exploring symbol calculus
• experiments with elliptic polylogarithms

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Contributing

Contributions are welcome. Possible contributions include

• new notebooks exploring identities
• numerical algorithms for MPLs
• symbolic implementations of shuffle/stuffle algebras
• visualization tools for polylogarithmic functions

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License

This project is licensed under the GNU General Public License v3.0 (GPL-3.0).

You may redistribute and modify this software under the terms of the GPL v3 as published by the Free Software Foundation.

See the LICENSE file for the full license text.