"Smyth formula for Mahler measures"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
imported>Pythagoras0 |
||
| 2번째 줄: | 2번째 줄: | ||
;thm '''[Smyth1981]''' | ;thm '''[Smyth1981]''' | ||
$$ | $$ | ||
| − | m(1+x_1+x_2)=L_{-3}'(-1)=\frac{3\sqrt{3}}{4\pi} | + | 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 | 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}== | ==two proofs of \ref{Smyth1}== | ||
2015년 2월 1일 (일) 18:15 판
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.