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


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.


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

Mathematics Subject Classification

11F67 33C20 



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].


  1. 1.
    Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, New York (1998) CrossRefMATHGoogle Scholar
  2. 2.
    Bertin, M.J.: Mahler’s measure and L-series of K3 hypersurfaces. In: Mirror Symmetry. V. AMS/IP Stud. Adv. Math., vol. 38, pp. 3–18. Amer. Math. Soc., Providence (2006) Google Scholar
  3. 3.
    Bertin, M.J.: Mesure de Mahler d’hypersurfaces K3. J. Number Theory 128, 2890–2913 (2008) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Boyd, D.W.: Mahler’s measure and special values of L-functions. Exp. Math. 7, 37–82 (1998) CrossRefMATHGoogle Scholar
  5. 5.
    Brillhart, J., Morton, P.: Table Erata: Heinrich Weber, 3rd edn. Lehrbuch der Algebra, vol. 3. Chelsea, New York (1961). Math. Comp. 65, 1379 (1996) Google Scholar
  6. 6.
    Deninger, C.: Deligne periods of mixed motives, K-theory and the entropy of certain \(\mathbb{Z}^{n}\)-actions. J. Am. Math. Soc. 10, 259–281 (1997) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dirichlet, P.G.L.: Vorlesungen über Zahlentheorie. Chelsea, New York (1968) Google Scholar
  8. 8.
    Dummit, D., Kisilevsky, H., McKay, J.: Multiplicative products of η-functions. In: Finite Groups—Coming of Age. Contemp. Math., vol. 45, pp. 89–98. Amer. Math. Soc., Providence (1985) CrossRefGoogle Scholar
  9. 9.
    Glasser, M.L., Zucker, I.J.: Lattice sums, Theoretical Chemistry—Advances and Perspectives. V, pp. 67–139. Academic Press, New York (1980) Google Scholar
  10. 10.
    Iwaniec, H.: Topics in Classical Automorphic Forms. Amer. Math. Soc., Providence (1997) MATHGoogle Scholar
  11. 11.
    Köhler, G.: Eta Products and Theta Series Identities. Springer, Berlin (2011) CrossRefMATHGoogle Scholar
  12. 12.
    Livné, R.: Motivic orthogonal two-dimensional representations of \(\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\). Isr. J. Math. 92, 149–156 (1995) CrossRefMATHGoogle Scholar
  13. 13.
    Ono, K.: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series. Amer. Math. Soc., Providence (2004) Google Scholar
  14. 14.
    Rodriguez Villegas, F.: In: Modular Mahler measures I. Topics in Number Theory, University Park, PA, 1997, pp. 17–48. Kluwer, Dordrecht (1999) Google Scholar
  15. 15.
    Rogers, M.D.: New 5 F 4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/π. Ramanujan J. 18, 327–340 (2009) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Rogers, M.D., Zudilin, W.: On the Mahler measure of 1+X+1/X+Y+1/Y. Int. Math. Res. Not. (2013). arXiv:1102.1153 [math.NT]
  17. 17.
    Schütt, M.: CM newforms with rational coefficients. Ramanujan J. 19, 187–205 (2009) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Smyth, C.J.: On measures of polynomials in several variables. Bull. Aust. Math. Soc. 23, 49–63 (1981) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Stein, E.M., Shakarchi, R.: Fourier Analysis: An Introduction. Princeton University Press, Princeton (2003) Google Scholar
  20. 20.
    Weber, H.: Lehrbuch der Algebra, vol. III. Vieweg, Braunschweig (1908) Google Scholar

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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