Abstract
In this paper we prove that the Mahler measures of the Laurent polynomials (x+x −1)(y+y −1)(z+z −1)+k 1/2, (x+x −1)2(y+y −1)2(1+z)3 z −2−k, and x 4+y 4+z 4+1+k 1/4 xyz, for various values of k, are of the form r 1 L′(f,0)+r 2 L′(χ,−1), where \(r_{1},r_{2}\in \mathbb{Q}\), f is a CM newform of weight 3, and χ is a quadratic character. Since it has been proved that these Mahler measures can also be expressed in terms of logarithms and 5 F 4-hypergeometric series, we obtain several new hypergeometric evaluations and transformations from these results.
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Acknowledgements
The author would like to thank Matthew Papanikolas for pointing out the numerical evidence of the first formula in Corollary 1.3, which chiefly inspires the author to write this paper, and many helpful discussions. The author is also grateful to Mathew Rogers for useful advice and suggestions. Finally, the author thanks Bruce Berndt for directing him to reference [5].
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The author’s research was partially supported by NSF Grant DMS-0903838.
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Samart, D. Three-variable Mahler measures and special values of modular and Dirichlet L-series. Ramanujan J 32, 245–268 (2013). https://doi.org/10.1007/s11139-013-9464-4
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DOI: https://doi.org/10.1007/s11139-013-9464-4
Keywords
- Mahler measures
- Eisenstein–Kronecker series
- Hecke L-series
- CM newforms
- L-functions
- Hypergeometric series