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Mahler measures, short walks and log-sine integrals. (English) Zbl 1297.11136

The authors investigate multiple Mahler measures that arise from a study of uniform random walks in the plane introduced by Pearson in 1905. The \(s\)th moment of the distance traveled by an \(n\)-step uniform random walk is given by \[ W_n(s)=\int_{0}^1 \dots \int_{0}^1 \Big|\sum_{k=1}^n e^{2\pi i t_k}\Big|^s dt_1 \dots dt_n. \] The authors review some earlier results on various values of \(W_n(s)\), their relation with hypergeometric functions and give some simple evaluation of some known Mahler measures of polynomials in several variables. They also evaluate some values of log-sine integrals \(Ls_{m}(2\pi)\) and \(Ls_m(\pi)\), defined by \[ -\frac{1}{\pi} \sum_{m=0}^{\infty} Ls_{m+1}(\pi) \frac{\lambda^m}{m!}={\lambda \choose \lambda/2}. \]

MSC:

11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11M32 Multiple Dirichlet series and zeta functions and multizeta values
33E20 Other functions defined by series and integrals
60G50 Sums of independent random variables; random walks
68W30 Symbolic computation and algebraic computation

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