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The Ramanujan Journal

, Volume 32, Issue 2, pp 245–268 | Cite as

Three-variable Mahler measures and special values of modular and Dirichlet L-series

  • Detchat Samart
Article

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 −2k, 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.

Keywords

Mahler measures Eisenstein–Kronecker series Hecke L-series CM newforms L-functions Hypergeometric series 

Mathematics Subject Classification

11F67 33C20 

Notes

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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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

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