Abstract
We prove an identity relating Mahler measures of a certain family of non-tempered polynomials to those of tempered polynomials. Evaluations of Mahler measures of some polynomials in the first family are also given in terms of special values of L-functions and logarithms.Finally, we prove Boyd’s conjectures for conductor 30 elliptic curves using our new identity, Brunault–Mellit–Zudilin’s formula and additional functional identities for Mahler measures.
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Acknowlegements
This research is supported financially by the Thailand Research Fund (TRF) Grant from the TRF and the Office of the Higher Education Commission, under the Contract No. MRG6280045. Part of this work was done during the second author’s visit to the Vietnam Institute for Advanced Study in Mathematics (VIASM) in the Summer semester of year 2018, partially supported under the institute’s research Contract No. NC/2018/VNCCCT. The second author is grateful to the VIASM for their support and hospitality. The authors would like to thank Bruce Berndt and Wadim Zudilin for their valuable comments on an earlier version of this paper. Lastly, the authors thank the anonymous referee for insightful remarks which greatly help to improve the exposition of this paper and for a suggestion about possible approach to completing the proof of Boyd’s conductor 30 conjectures.
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Meemark, Y., Samart, D. Mahler measures of a family of non-tempered polynomials and Boyd’s conjectures. Res Math Sci 7, 1 (2020). https://doi.org/10.1007/s40687-019-0200-6
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DOI: https://doi.org/10.1007/s40687-019-0200-6