Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 3
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Borwein, Jonathan M. Straub, Armin and Vignat, Christophe 2016. Densities of short uniform random walks in higher dimensions. Journal of Mathematical Analysis and Applications, Vol. 437, Issue. 1, p. 668.

    Pardo, Luis M. and Pardo, Mario 2016. On the zeta Mahler measure function of the Jacobian determinant, condition numbers and the height of the generic discriminant. Applicable Algebra in Engineering, Communication and Computing, Vol. 27, Issue. 4, p. 303.

    Borwein, Jonathan M. and Straub, Armin 2013. Mahler measures, short walks and log-sine integrals. Theoretical Computer Science, Vol. 479, p. 4.

  • Currently known as: Journal of the Australian Mathematical Society Title history
    Journal of the Australian Mathematical Society, Volume 92, Issue 1
  • February 2012, pp. 15-36


  • DOI:
  • Published online: 15 June 2012

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.

Corresponding author
For correspondence; e-mail:
Hide All

Borwein was supported in part by the Australian Research Council and the University of Newcastle.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]H. Akatsuka , ‘Zeta Mahler measures’, J. Number Theory 129(11) (2009), 27132734.

[4]J. M. Borwein , D. M. Bradley , D. J. Broadhurst and P. Lisoněk , ‘Special values of multiple polylogarithms’, Trans. Amer. Math. Soc. 353(3) (2001), 907941.

[5]J. M. Borwein , D. J. Broadhurst and J. Kamnitzer , ‘Central binomial sums, multiple Clausen values, and zeta values’, Experiment. Math. 10(1) (2001), 2534.

[6]J. M. Borwein , O.-Y. Chan and R. E. Crandall , ‘Higher-dimensional box integrals’, Experiment. Math. 19(3) (2010), 431446.

[10]J. M. Borwein , I. J. Zucker and J. Boersma , ‘The evaluation of character Euler double sums’, Ramanujan J. 15(3) (2008), 377405.

[11]D. W. Boyd , ‘Speculations concerning the range of Mahler’s measure’, Canad. Math. Bull. 24 (1981), 453469.

[12]A. Davydychev and M. Kalmykov , ‘New results for the ε-expansion of certain one-, two- and three-loop Feynman diagrams’, Nuclear Phys. B 605 (2001), 266318.

[15]M. Kalmykov , ‘About higher order ϵ-expansion of some massive two- and three-loop master-integrals’, Nuclear Phys. B 718 (2005), 276292.

[16]N. Kurokawa , M. Lalín and H. Ochiai , ‘Higher Mahler measures and zeta functions’, Acta Arith. 135(3) (2008), 269297.

[17]M. Lalín and K. Sinha , ‘Higher Mahler measure of cyclotomic polynomials and Lehmer’s question’, Ramanujan J. 26 (2011), 257294.

[19]L. Lewin , ‘On the evaluation of log-sine integrals’, The Mathematical Gazette 42 (1958), 125128.

[21]A. v.  d. Poorten , ‘A proof that Euler missed…Apéry’s proof of the irrationality of ζ(3)’, Math. Intelligencer 1(4) (1979), 195203.

[22]Y. Sasaki , ‘On multiple higher Mahler measures and multiple L values’, Acta Arith. 144(2) (2010), 159165.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *