"다중 제타 값"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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| 51번째 줄: | 51번째 줄: | ||
==메모== | ==메모== | ||
* http://math.unice.fr/~brunov/GdT/The%20Algebra%20Of%20Multiple%20Zeta%20Values.pdf | * http://math.unice.fr/~brunov/GdT/The%20Algebra%20Of%20Multiple%20Zeta%20Values.pdf | ||
| + | * [http://www.usna.edu/Users/math/meh/biblio.html REFERENCES ON MULTIPLE ZETA VALUES AND EULER SUMS] | ||
| 65번째 줄: | 66번째 줄: | ||
==사전 형태의 자료== | ==사전 형태의 자료== | ||
* http://en.wikipedia.org/wiki/Multiple_zeta_function | * http://en.wikipedia.org/wiki/Multiple_zeta_function | ||
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| + | ==리뷰, 에세이, 강의노트== | ||
| + | * Brown, Francis. “Motivic Periods and the Projective Line Minus Three Points.” arXiv:1407.5165 [math], July 19, 2014. http://arxiv.org/abs/1407.5165. | ||
| + | * http://www.math.jussieu.fr/~leila/MIT4B.pdf | ||
| + | * [http://www.maths.bris.ac.uk/events/meetings/uploads/4159LeilaSchnepsII.pdf Survey of the theory of multiple zeta values], Leila Schneps | ||
| + | * Pierre Deligne ''Multizetas, d´aprés Francis Brown'', Seminaire Bourbaki, Nr. 1048, Januar 2012, http://www.math.ias.edu/files/deligne/012312MultiZetas.pdf | ||
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| 70번째 줄: | 80번째 줄: | ||
* Zudilin, Wadim. ‘On a Family of Polynomials Related to $\zeta(2,1)=\zeta(3)$’. arXiv:1504.07696 [math-Ph], 28 April 2015. http://arxiv.org/abs/1504.07696. | * Zudilin, Wadim. ‘On a Family of Polynomials Related to $\zeta(2,1)=\zeta(3)$’. arXiv:1504.07696 [math-Ph], 28 April 2015. http://arxiv.org/abs/1504.07696. | ||
* Furusho, Hidekazu. “On Relations among Multiple Zeta Values Obtained in Knot Theory.” arXiv:1501.06638 [math], January 26, 2015. http://arxiv.org/abs/1501.06638. | * Furusho, Hidekazu. “On Relations among Multiple Zeta Values Obtained in Knot Theory.” arXiv:1501.06638 [math], January 26, 2015. http://arxiv.org/abs/1501.06638. | ||
| + | * Brown, Francis. “Multiple Modular Values for SL_2(Z).” arXiv:1407.5167 [math], July 19, 2014. http://arxiv.org/abs/1407.5167. | ||
* Zhao, Jianqiang. “Uniform Approach to Double Shuffle and Duality Relations of Various Q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras.” arXiv:1412.8044 [math], December 27, 2014. http://arxiv.org/abs/1412.8044. | * Zhao, Jianqiang. “Uniform Approach to Double Shuffle and Duality Relations of Various Q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras.” arXiv:1412.8044 [math], December 27, 2014. http://arxiv.org/abs/1412.8044. | ||
* Blümlein, J., D. J. Broadhurst, and J. A. M. Vermaseren. ‘The Multiple Zeta Value Data Mine’. Computer Physics Communications 181, no. 3 (March 2010): 582–625. doi:10.1016/j.cpc.2009.11.007. | * Blümlein, J., D. J. Broadhurst, and J. A. M. Vermaseren. ‘The Multiple Zeta Value Data Mine’. Computer Physics Communications 181, no. 3 (March 2010): 582–625. doi:10.1016/j.cpc.2009.11.007. | ||
2015년 5월 1일 (금) 00:53 판
개요
- 리만제타함수의 다변수 일반화 $\zeta(s_1, \ldots, s_k)$
- $s_i$ 가 양의 정수일 때, 오일러 합이라 불림
- 정수론의 중요한 주제로 물리에서 산란 amplitude 등의 계산에서 등장
정의
- 다중 제타 값을 다음과 같이 정의
\[ \zeta(s_1, \ldots, s_k) : = \sum_{n_1 > n_2 > \cdots > n_k > 0} \ \frac{1}{n_1^{s_1} \cdots n_k^{s_k}} = \sum_{n_1 > n_2 > \cdots > n_k > 0} \ \prod_{i=1}^k \frac{1}{n_i^{s_i}}, \!\]
- $s_1, \ldots, s_k$가 정수일 때, $w=s_1+\cdots+s_k$를 weight, $k$를 depth로 부른다
이중 제타
- 오일러의 공식
$$\zeta(2,1)=\zeta(3)$$ $$\zeta(1,2)=-2\zeta(3)$$
여러 가지 관계식
double shuffle
- 정리
$m,n>1$ 일 때, $$\zeta(m)\zeta(n)=\zeta(m,n)+\zeta(n,m)+\zeta(m+n)$$
- 증명
$$ \zeta(m)\zeta(n)=(\sum_{j}\frac{1}{j^{m}})(\sum_{k}\frac{1}{k^{n}})=\sum_{j>k}\frac{1}{j^mk^n}+\sum_{j=k}\frac{1}{j^mk^n}+\sum_{j<k}\frac{1}{j^mk^n} $$
오일러 분해 공식
- $r,s>1$ 일 때,
$$\zeta(r)\zeta(s)=\sum_{a=0}^{s-1}\binom{a+r-1}{a}\zeta(r+a,s-a)+\sum_{a=0}^{r-1}\binom{a+s-1}{a}\zeta(s+a,r-a)$$
기타
- 다음이 성립한다
$$ 2\zeta(n,1)=n \zeta(n+1)-\sum_{i=1}^{a-2}\zeta(n-i)\zeta(i+1) $$
- 예
$$\zeta(2,1)=\zeta(3)$$ $$\zeta(4,1)=2\zeta(5)-\zeta(2)\zeta(3)$$
일차독립인 다중 제타 값의 공간
- 주어진 무게를 갖는 다중 제타 값이 이루는 벡터 공간의 차원
- $\{a_n\}_{n=1}^{\infty}$를 $a_n = a_{n-2} + a_{n-3}$, $a_0=1, a_1=a_2=0$.
- 이를 파도반 수열이라 하자
- 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897
- Zagier conjectures that a(n+3) is the maximum number of multiple zeta values of weight n > 1 which are linearly independent over the rationals.
메모
- http://math.unice.fr/~brunov/GdT/The%20Algebra%20Of%20Multiple%20Zeta%20Values.pdf
- REFERENCES ON MULTIPLE ZETA VALUES AND EULER SUMS
관련된 항목들
계산 리소스
사전 형태의 자료
리뷰, 에세이, 강의노트
- Brown, Francis. “Motivic Periods and the Projective Line Minus Three Points.” arXiv:1407.5165 [math], July 19, 2014. http://arxiv.org/abs/1407.5165.
- http://www.math.jussieu.fr/~leila/MIT4B.pdf
- Survey of the theory of multiple zeta values, Leila Schneps
- Pierre Deligne Multizetas, d´aprés Francis Brown, Seminaire Bourbaki, Nr. 1048, Januar 2012, http://www.math.ias.edu/files/deligne/012312MultiZetas.pdf
관련논문
- Zudilin, Wadim. ‘On a Family of Polynomials Related to $\zeta(2,1)=\zeta(3)$’. arXiv:1504.07696 [math-Ph], 28 April 2015. http://arxiv.org/abs/1504.07696.
- Furusho, Hidekazu. “On Relations among Multiple Zeta Values Obtained in Knot Theory.” arXiv:1501.06638 [math], January 26, 2015. http://arxiv.org/abs/1501.06638.
- Brown, Francis. “Multiple Modular Values for SL_2(Z).” arXiv:1407.5167 [math], July 19, 2014. http://arxiv.org/abs/1407.5167.
- Zhao, Jianqiang. “Uniform Approach to Double Shuffle and Duality Relations of Various Q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras.” arXiv:1412.8044 [math], December 27, 2014. http://arxiv.org/abs/1412.8044.
- Blümlein, J., D. J. Broadhurst, and J. A. M. Vermaseren. ‘The Multiple Zeta Value Data Mine’. Computer Physics Communications 181, no. 3 (March 2010): 582–625. doi:10.1016/j.cpc.2009.11.007.