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Log-sine evaluations of Mahler measures. II. (English) Zbl 1266.33022

The aim of this paper is to apply properties of log-sine integrals to the study of higher and multiple Mahler measures as a continuation of previous work [the second and the third author, J. Aust. Math. Soc. 92, No. 1, 15–36 (2012; Zbl 1277.33019); “Special values of generalized log-sine integrals”, ISSAC 2011. Proceedings of the International Symposium on Symbolic and Algebraic Computation, New York: ACM Press, 43–50 (2011; Zbl 1271.68047), arXiv:1103.4298].
The multiple Mahler measure of \(n\)-variable polynomials \(P_1, \dotsc, P_k\) was given in [N. Kurokawa et al., Acta Arith. 135, No. 3, 269–297 (2008; Zbl 1211.11116)] by \[ \mu(P_1,\dotsc, P_k)\;:=\int_0^1 \dotsm\int_0^1 \log|P_1(e^{2\pi i \theta_1},\dotsc, e^{2\pi i \theta_n})|\dotsm \log|P_k(e^{2\pi i \theta_1},\dotsc, e^{2\pi i \theta_n})|d\theta_1\dotsm d \theta_n, \] and the higher Mahler measure corresponds to the particular case in which \(P_1=\dotsb=P_k\).
The authors discuss some formulas for \(\mu(1-x,\dotsc, 1-x, 1+x, \dotsc, 1+x)\) and \(\mu_k(1+x+y)\), in relation to log-sine integrals, and more importantly, to polylogarithms.

MSC:

33E20 Other functions defined by series and integrals
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11R09 Polynomials (irreducibility, etc.)

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