"Smyth formula for Mahler measures"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
;thm '''[Smyth1981]''' | ;thm '''[Smyth1981]''' | ||
| − | + | :<math> | |
m(1+x_1+x_2)=L_{-3}'(-1)=\frac{3\sqrt{3}}{4\pi}L(\chi_{-3},2)=0.3230659472\cdots \label{Smyth1} | m(1+x_1+x_2)=L_{-3}'(-1)=\frac{3\sqrt{3}}{4\pi}L(\chi_{-3},2)=0.3230659472\cdots \label{Smyth1} | ||
| − | + | </math> | |
where | where | ||
| − | + | :<math>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</math> | |
| − | + | :<math> | |
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 | ||
| − | + | </math> | |
==two proofs of \ref{Smyth1}== | ==two proofs of \ref{Smyth1}== | ||
2020년 11월 13일 (금) 23:45 기준 최신판
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.