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Higher Mahler measure for cyclotomic polynomials and Lehmer’s question

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Abstract

The k-higher Mahler measure of a non-zero polynomial P is the integral of log k|P| on the unit circle. In this note, we consider Lehmer’s question (which is a long-standing open problem for k=1) for k>1 and find some interesting formulas for 2- and 3-higher Mahler measure of cyclotomic polynomials.

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Authors and Affiliations

  1. Département de mathématiques et de statistique, Université de Montréal, Montréal, QC, H3C3J7, Canada

    Matilde Lalín

  2. Department of Mathematical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur, Nadia, 741252, West Bengal, India

    Kaneenika Sinha

Authors
  1. Matilde Lalín
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  2. Kaneenika Sinha
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Correspondence to Matilde Lalín.

Additional information

M.L. was supported by NSERC Discovery Grant 355412-2008 and a Faculty of Science Startup grant from the University of Alberta. K.S. was supported by a Pacific Institute for the Mathematical Sciences Postdoctoral Fellowship and the above grants.

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Lalín, M., Sinha, K. Higher Mahler measure for cyclotomic polynomials and Lehmer’s question. Ramanujan J 26, 257–294 (2011). https://doi.org/10.1007/s11139-010-9278-6

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  • DOI: https://doi.org/10.1007/s11139-010-9278-6

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