The Mahler measure of a three-variable family and an application to the Boyd–Lawton formula


We prove a formula relating the Mahler measure of an infinite family of three-variable polynomials to a combination of the Riemann zeta function at \(s=3\) and special values of the Bloch–Wigner dilogarithm by evaluating a regulator. The evaluation requires two different applications of Jensen’s formula and analyzing the integral in two different planes, as opposed to the more common strategy of using only one plane. The degrees of the monomials involving one of the variables are allowed to vary freely, leading to an interesting application of the Boyd–Lawton formula.

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The authors are grateful to David Boyd and Mathew Rogers for some early discussions in the topic of these formulas and would like to thank the referee for helpful suggestions and corrections.

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Correspondence to Jarry Gu.

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This work was supported by the Natural Sciences and Engineering Research Council of Canada [Discovery Grant 355412-2013], the Fonds de recherche du Québec - Nature et technologies [Projet de recherche en équipe 256442], and the Institut des sciences mathématiques.

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Gu, J., Lalín, M. The Mahler measure of a three-variable family and an application to the Boyd–Lawton formula. Res. number theory 7, 13 (2021).

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  • Mahler measure
  • Boyd–Lawton formula
  • Regulator

Mathematics Subject Classification

  • Primary 11R06
  • Secondary 11M06
  • 11R42