Mahler measure

introduction

• for a Laurent polynomial $P(x_1,\cdots, x_n)\in \mathbb{Z}[x_1^{\pm 1},\cdots,x_n^{\pm 1}]$, the Mahler measure is defined to be

$$ m(P):=\int_{0}^{1}\cdots \int_{0}^{1} \ln |P(e^{2\pi i \theta_1},\cdots, e^{2\pi i \theta_n})|\, d\theta_1\cdots d\theta_n $$

examples

Smyth

• 1981

$$ m(1+x_1+x_2)=L_{-3}'(-1)=\frac{3\sqrt{3}}{4\pi}L_{-3}(2)=0.3230659472\cdots $$

$$ m(1+x_1+x_2+x_3)=14\zeta'(-2)=\frac{7}{2\pi^2}\zeta(3)=0.4262783988\cdots $$

Rodriguez-Villegas

$$ m(1+x_1+x_2+x_3+x_4)=-L_{f}'(-1)=\frac{675\sqrt{15}}{16\pi^5}L_{f}(4)=0.5444125617\cdots $$

• A conjecture due to F. Rodriguez-Villegas represents this Mahler measure as a special value at the point 4 of the L-function of a modular modular form of weight 3. We prove that this Mahler measure is equal to a linear combination of double L-values of certain meromorphic modular forms of weight 4.

$$ m(1+x_1+x_2+x_3+x_4+x_5)=-8L_{g}'(-1)=\frac{648}{\pi^6}L_{g}(5)=0.6273170748\cdots $$ where $$ f(\tau)=\eta(3\tau)^3\eta(5\tau)^3+\eta(\tau)^3\eta(15\tau)^3 $$ and $$ g(\tau)=\eta(\tau)^2\eta(2\tau)^2\eta(3\tau)^2\eta(6\tau)^2 $$

Deninger

$$ m(1+x+\frac{1}{x}+y+\frac{1}{y})=L_{15A}'(0)=\frac{15}{4\pi^2}L_{15A}(2)=0.2513304337\cdots $$

Boyd

$$ m(1+x+\frac{1}{x}+y+\frac{1}{y}+xy+\frac{1}{xy})=L_{14A}'(0)=\frac{7}{2\pi^2}L_{14A}(2)=0.2274812230\cdots $$

$$ m(-1+x+\frac{1}{x}+y+\frac{1}{y}+xy+\frac{1}{xy})=L_{30A}'(0)=\frac{15}{2\pi^2}L_{30A}(2)=0.6168709387\cdots $$

• Boyd conjecture on Mahler measure of three variables polynomial
• Mahler measures and L-values of elliptic curves

related items

• Zagier's conjecture
• A log tangent integral

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxSTAxSnVRUmpEeFU/edit
• http://mathworld.wolfram.com/LehmersMahlerMeasureProblem.html
• http://mathworld.wolfram.com/MahlerMeasure.html

encyclopedia

• http://en.wikipedia.org/wiki/Mahler_measure

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
• Lalin Mahler measures as values of regulators 2006
• Finch, Modular Forms on $SL_2(\mathbb{Z})$ 2005
• The many aspects of Mahler's measure, Banff workshop, 2003
• Lalin Introduction to Mahler measure, 2003
• Boyd, Mahler's measure, hyperbolic geometry and the dilogarithm 2002
• Boyd, David W. 1998. “Mahler’s Measure and Special Values of $L$-Functions.” Experimental Mathematics 7 (1): 37–82.

lecture notes

• Course at Harvard University Spring 2002.
  
• Fernando Rodriguez Villegashttp://www.ma.utexas.edu/users/villegas/KL/
  
• Suggested exercises on periods.
  
• Notes last updated: May 14, 2002.
  
• The following are class notes taken by Sam Vandervelde (samv@mandelbrot.org). Please let me or Sam know of any comments, corrections, etc. Thanks.
  
• sam-notes-1-1.pdf
• 3313085/attachments/2687571 sam-notes-2.pdf
• sam-notes-3.pdf
• sam-notes-4.pdf
• sam-notes-5.pdf
• Notes 1
• Notes 2
• Notes 3
• Notes 4
• Notes 5
• The following are class notes taken by Matilde Lalin (mlalin@math.harvard.edu). Please let me or Matilde know of any comments, corrections, etc. Thanks.
  
• Notes 6

articles

• Zudilin, Wadim. 2013. “Regulator of Modular Units and Mahler Measures”. ArXiv e-print 1304.3869. http://arxiv.org/abs/1304.3869.
• Shinder, Evgeny, and Masha Vlasenko. 2012. “Linear Mahler Measures and Double L-Values of Modular Forms.” arXiv:1206.1454 [math]. http://arxiv.org/abs/1206.1454.
• A dynamical interpretation of the global canonical height on an elliptic curve
• Mahler's Measure and the Dilogarithm (I)
• Mahler's Measure and the Dilogarithm (II)
  • Authors: David W. Boyd, Fernando Rodriguez-Villegas, Nathan M. Dunfield
• C. J. Smyth, An explicit formula for the Mahler measure of a family of 3-variable polynomials, J. Th. Nombres Bordeaux 14 (2002), 683{700
• 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.

links

• http://enyokoyama.blogspot.com/2012/05/mahlers-measure-hyperbolic-geometry-and.html