Mahler measures, short walks and log-sine integrals

Abstract

The Mahler measure of a polynomial in several variables has been a subject of much study over the past thirty years -very few closed forms are proven but more are conjectured. In the case of multiple Mahler measures more tractable but interesting families exist. Using values of log-sine integrals we provide systematic evaluations of various higher and multiple Mahler measures.The evaluations in terms of log-sine integrals become particularly useful in light of the fact that log-sine integrals may be automatically reexpressed as polylogarithmic values. We present this correspondence along with related generating functions for log-sine integrals.Our initial interest in considering Mahler measures stems from a study of uniform random walks in the plane as first introduced by Pearson. The main results on the moments of the distance traveled by an n-step walk, as well as the corresponding probability density functions, are reviewed. It is the derivative values of the moments that are Mahler measures.This work would be impossible without very extensive symbolic and numeric computations. It also makes frequent use of the new NIST Handbook of Mathematical Functions and similar tools. Our intention is to show off the interplay between numeric and symbolic computing while exploring the three mathematical topics in the title.

Relations for Nielsen polylogarithms

“…Here, as elsewhere, the chances of introducing error without using a computer algebra system to verify identity(9) are very high and such errors pervade the literature.…”

“…In this note, we consider the entirety of these relations. Besides generic relations as above, we will be interested in additional relations at the special arguments π/3, π/2 and 2π/3 because of their appearance in applications, see [8], both within mathematics (e.g., Apéry-like series for zeta values, multiple Mahler measure [9]) and physics (e.g., calculation of higher terms in ε-expansions of Feynman diagrams [13,15]). We should explicitly mention Broadhurst whose extensive work in [10] on more general polylogarithms in the sixth root of unity, see (29), is indicated in Remarks 3.9 and 4.2.…”

Experimental Computation and Visual Theorems

“…6 Proteins participate in virtually every cellular process and Protein structure → predicts how functions are performed. NMR spectroscopy (Nuclear Overhauser effect 7 ) can determine a subset of interatomic distances without damage (under 6Å ).…”

“…Our group, motivate initially by multi-dimensional quadrature techniques for higher precision than Monte Carlo can provide looked at the moments and densities of n-step walks of unit size with uniform random angles [6,8]. Intensive numeric-symbolic and graphic computing lead to some striking new results for a century old problem.…”

“…(See Case studies I and II.) I end with description of recent work from my group in probability (behaviour of short random walks [6,8]) and transcendental number theory (normality of real numbers [2] …”