Log-sine evaluations of Mahler measures, II

We continue the analysis of higher and multiple Mahler measures using log-sine integrals as started in "Log-sine evaluations of Mahler measures" and "Special values of generalized log-sine integrals" by two of the authors. This motivates a detailed study of various multiple polylogarithms and worked examples are given. Our techniques enable the reduction of several multiple Mahler measures, and supply an easy proof of two conjectures by Boyd.Comment: 35 page

Log-sine evaluations of Mahler measures

We provide evaluations of several recently studied higher and multiple Mahler measures using log-sine integrals. This is complemented with an analysis of generating functions and identities for log-sine integrals which allows the evaluations to be expressed in terms of zeta values or more general polylogarithmic terms. The machinery developed is then applied to evaluation of further families of multiple Mahler measures.Comment: 25 page

多重ゼータ値とlog-sine積分について (解析的整数論とその周辺)

本稿では反復log-sine積分を導入しその基本的な性質を述べた後, 反復log-sine積分を使って多重ゼータ値の間の関係式を得る方法を紹介する

Special values of generalized log-sine integrals

We study generalized log-sine integrals at special values. At π and multiples thereof explicit evaluations are obtained in terms of multiple polylogarithms at ±1. For general arguments we present algorithmic evaluations involving polylogarithms at related arguments. In particular, we consider log-sine integrals at π/3 which evaluate in terms of polylogarithms at the sixth root of unity. An implementation of our results for the computer algebra systems Mathematica and SAGE is provided. 1