Smyth formula for Mahler measures
imported>Pythagoras0님의 2015년 1월 25일 (일) 00:19 판 (새 문서: ==introduction== ;thm '''[Smyth1981]''' $$ m(1+x_1+x_2)=L_{-3}'(-1)=\frac{3\sqrt{3}}{4\pi}L_{-3}(2)=0.3230659472\cdots \label{Smyth1} $$ $$ m(1+x_1+x_2+x_3)=14\zeta'(-2)=\frac{7}{2\p...)
introduction
- thm [Smyth1981]
$$ m(1+x_1+x_2)=L_{-3}'(-1)=\frac{3\sqrt{3}}{4\pi}L_{-3}(2)=0.3230659472\cdots \label{Smyth1} $$
$$ 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
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.