Smyth formula for Mahler measures

introduction

thm [Smyth1981]

$$ m(1+x_1+x_2)=L_{-3}'(-1)=\frac{3\sqrt{3}}{4\pi}L(\chi_{-3},2)=0.3230659472\cdots \label{Smyth1} $$ where $$L(\chi_{-3},s)=\sum_{n=1}^{\infty}\frac{\chi_{-3}(n)}{n^s}=\frac{1}{1^s}-\frac{1}{2^s}+\frac{1}{4^s}-\frac{1}{5^s}+\cdots$$ $$ m(1+x_1+x_2+x_3)=14\zeta'(-2)=\frac{7}{2\pi^2}\zeta(3)=0.4262783988\cdots $$

two proofs of \ref{Smyth1}

• direct calculation
• using regulator

expositions

• Smyth, Chris. 2008. “The Mahler Measure of Algebraic Numbers: a Survey.” In Number Theory and Polynomials, 352:322–349. London Math. Soc. Lecture Note Ser. Cambridge: Cambridge Univ. Press. http://www.maths.ed.ac.uk/~chris/Smyth240707.pdf http://arxiv.org/pdf/math/0701397.pdf

articles

• [Smyth1981] Smyth, C. J. 1981. “On Measures of Polynomials in Several Variables.” Bulletin of the Australian Mathematical Society 23 (1): 49–63. doi:http://dx.doi.org/10.1017/S0004972700006894.