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March  2020, 28(1): 103-125. doi: 10.3934/era.2020007

The Mahler measure of $ (x+1/x)(y+1/y)(z+1/z)+\sqrt{k} $

Department of Mathematics, Nanjing Univeristy, Nanjing, Jiangsu 210093, China

* Corresponding author: Hourong Qin

Received  October 2019 Revised  December 2019 Published  March 2020

Fund Project: The authors are supported by National Nature Science Foundation of China (Nos. 11571163, 11631009).

In this paper we study the Mahler measures of reciprocal polynomials $ (x+1/x)(y+1/y)(z+1/z)+\sqrt{k} $ for $ k = 16 $, $ k = -104\pm60\sqrt{3} $, $ 4096 $ and $ k = -2024\pm765\sqrt{7} $. We prove six conjectural identities proposed by Samart in [16].

Citation: Huimin Zheng, Xuejun Guo, Hourong Qin. The Mahler measure of $ (x+1/x)(y+1/y)(z+1/z)+\sqrt{k} $. Electronic Research Archive, 2020, 28 (1) : 103-125. doi: 10.3934/era.2020007
References:
[1]

M. J. Bertin, Mahler's measure and $L$-series of $K3$ hypersurfaces, Mirror Symmetry, AMS/IP Stud. Adv. Math., Amer. Math. Soc. Providence, RI, 38 (2006), 3–18. Google Scholar

[2]

M. J. Bertin, Mesure de Mahler d'hypersurfaces $K3$, J. Number Theory, 128 (2008), 2890-2913.  doi: 10.1016/j.jnt.2007.12.012.  Google Scholar

[3]

B. J. Birch, Weber's class invariants, Mathematika, 16 (1969), 283-294.  doi: 10.1112/S0025579300008251.  Google Scholar

[4]

D. W. Boyd, Mahler's measure and special values of $L$-functions, Experimental Math., 7 (1998), 37-82.  doi: 10.1080/10586458.1998.10504357.  Google Scholar

[5]

C. Deninger, Deligne periods of mixed motives, K-theory and the entropy of certain $\mathbb{Z}_n$-actions, J. Amer. Math. Soc., 10 (1997), 259-281.  doi: 10.1090/S0894-0347-97-00228-2.  Google Scholar

[6] M. L. Glasser and I. J. Zucker, Lattice sums, Theoretical Chemistry - Advances and Perspectives V , Academic Press, New York, 1980.  doi: 10.1016/B978-0-12-681905-2.50008-6.  Google Scholar
[7]

X. Guo, Y. Peng and H. Qin, Three-variable Mahler measures and special values of $L$-functions of modular forms, Ramanujan J.. Google Scholar

[8]

M. N. Lalin and M. D. Rogers, Functional equations for Mahler measures of genus-one curves, Algebraic Number Theory, 1 (2007), 87-117.  doi: 10.2140/ant.2007.1.87.  Google Scholar

[9]

K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and Q-series, , Amer.Math. Soc., Providence, RI, 2004.  Google Scholar

[10]

M. Rogers, New ${}_5F_4$ hypergeometric transformations, three-variable Mahler measures, and formulas for $1/\pi$, Ramanujan J., 18 (2009), 327-340.  doi: 10.1007/s11139-007-9040-x.  Google Scholar

[11]

M. Rogers and W. Zudilin, From $L$-series of elliptic curves to Mahler measures, Compositio Math., 148 (2012), 385–414. (MR 2904192) doi: 10.1112/S0010437X11007342.  Google Scholar

[12]

M. Rogers and W. Zudilin, On the Mahler measure of $1+X +1/X +Y +1/Y$, Intern. Math. Res. Not., 2014 (2014), 2305-2326.  doi: 10.1093/imrn/rns285.  Google Scholar

[13]

The Sage Developers, SageMath, the Sage Mathematics Software System (Version 8.6), 2019, http://www.sagemath.org. Google Scholar

[14]

F. Rodriguez-Villegas, Modular Mahler Measures I, Topics in number theory (University Park, PA, 1997), 1748, Math. Appl., 467, Kluwer Acad. Publ., Dordrecht, 1999. Google Scholar

[15]

D. Samart, Three-variable Mahler measures and special values of modular and Dirichlet $L$-series, Ramanujan J., 32 (2013), 245-268.  doi: 10.1007/s11139-013-9464-4.  Google Scholar

[16]

D. Samart, Mahler measures as linear combinations of $L$-values of multiple modular forms, Canad. J. Math., 67 (2015), 424-449.  doi: 10.4153/CJM-2014-012-8.  Google Scholar

[17]

C. J. Smyth, On measures of polynomials in several variables, Bull. Austral. Math. Soc., 23 (1981), 49-63.  doi: 10.1017/S0004972700006894.  Google Scholar

[18]

H. Weber, Lehrbuch der Algebra, Bd. III, F. Vieweg & Sohn, Braunschweig, 1908. Google Scholar

[19]

N. Yui and D. Zagier, On the singular values of Weber modular functions, Math. Comp., 66 (1997), 1645-1662.  doi: 10.1090/S0025-5718-97-00854-5.  Google Scholar

show all references

References:
[1]

M. J. Bertin, Mahler's measure and $L$-series of $K3$ hypersurfaces, Mirror Symmetry, AMS/IP Stud. Adv. Math., Amer. Math. Soc. Providence, RI, 38 (2006), 3–18. Google Scholar

[2]

M. J. Bertin, Mesure de Mahler d'hypersurfaces $K3$, J. Number Theory, 128 (2008), 2890-2913.  doi: 10.1016/j.jnt.2007.12.012.  Google Scholar

[3]

B. J. Birch, Weber's class invariants, Mathematika, 16 (1969), 283-294.  doi: 10.1112/S0025579300008251.  Google Scholar

[4]

D. W. Boyd, Mahler's measure and special values of $L$-functions, Experimental Math., 7 (1998), 37-82.  doi: 10.1080/10586458.1998.10504357.  Google Scholar

[5]

C. Deninger, Deligne periods of mixed motives, K-theory and the entropy of certain $\mathbb{Z}_n$-actions, J. Amer. Math. Soc., 10 (1997), 259-281.  doi: 10.1090/S0894-0347-97-00228-2.  Google Scholar

[6] M. L. Glasser and I. J. Zucker, Lattice sums, Theoretical Chemistry - Advances and Perspectives V , Academic Press, New York, 1980.  doi: 10.1016/B978-0-12-681905-2.50008-6.  Google Scholar
[7]

X. Guo, Y. Peng and H. Qin, Three-variable Mahler measures and special values of $L$-functions of modular forms, Ramanujan J.. Google Scholar

[8]

M. N. Lalin and M. D. Rogers, Functional equations for Mahler measures of genus-one curves, Algebraic Number Theory, 1 (2007), 87-117.  doi: 10.2140/ant.2007.1.87.  Google Scholar

[9]

K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and Q-series, , Amer.Math. Soc., Providence, RI, 2004.  Google Scholar

[10]

M. Rogers, New ${}_5F_4$ hypergeometric transformations, three-variable Mahler measures, and formulas for $1/\pi$, Ramanujan J., 18 (2009), 327-340.  doi: 10.1007/s11139-007-9040-x.  Google Scholar

[11]

M. Rogers and W. Zudilin, From $L$-series of elliptic curves to Mahler measures, Compositio Math., 148 (2012), 385–414. (MR 2904192) doi: 10.1112/S0010437X11007342.  Google Scholar

[12]

M. Rogers and W. Zudilin, On the Mahler measure of $1+X +1/X +Y +1/Y$, Intern. Math. Res. Not., 2014 (2014), 2305-2326.  doi: 10.1093/imrn/rns285.  Google Scholar

[13]

The Sage Developers, SageMath, the Sage Mathematics Software System (Version 8.6), 2019, http://www.sagemath.org. Google Scholar

[14]

F. Rodriguez-Villegas, Modular Mahler Measures I, Topics in number theory (University Park, PA, 1997), 1748, Math. Appl., 467, Kluwer Acad. Publ., Dordrecht, 1999. Google Scholar

[15]

D. Samart, Three-variable Mahler measures and special values of modular and Dirichlet $L$-series, Ramanujan J., 32 (2013), 245-268.  doi: 10.1007/s11139-013-9464-4.  Google Scholar

[16]

D. Samart, Mahler measures as linear combinations of $L$-values of multiple modular forms, Canad. J. Math., 67 (2015), 424-449.  doi: 10.4153/CJM-2014-012-8.  Google Scholar

[17]

C. J. Smyth, On measures of polynomials in several variables, Bull. Austral. Math. Soc., 23 (1981), 49-63.  doi: 10.1017/S0004972700006894.  Google Scholar

[18]

H. Weber, Lehrbuch der Algebra, Bd. III, F. Vieweg & Sohn, Braunschweig, 1908. Google Scholar

[19]

N. Yui and D. Zagier, On the singular values of Weber modular functions, Math. Comp., 66 (1997), 1645-1662.  doi: 10.1090/S0025-5718-97-00854-5.  Google Scholar

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