Special functions, transcendentals and their numerics

TL;DR

This paper surveys cyclotomic polylogarithms and their numerical evaluation, focusing on constants that arise at the argument 1 and on PSLQ relations among these constants. It defines cyclotomic polylogarithms using cyclotomic polynomials, links them to harmonic and generalized polylogarithms, and discusses numerical strategies including series expansions and argument transforms. The authors prove all PSLQ relations for weights 1 and 2 from classical polylogarithm identities and show that cyclotomy 12 yields new constants beyond prior work, with analytic proofs partly established. They conclude that for k up to 6 and weight up to 6, all PSLQ relations are known, while cyclotomy 12 reveals new relations awaiting full treatment.

Abstract

Cyclotomic polylogarithms are reviewed and new results concerning the special constants that occur are presented. This also allows some comments on previous literature results using PSLQ.