다중 제타 값

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

$s_1\geq 1, \ldots, s_k$가 양의 정수라 하자
다중 제타 값을 다음과 같이 정의

\[ \zeta(s_1, \ldots, s_k) : = \sum_{n_1 > n_2 > \cdots > n_k > 0} \ \frac{1}{n_1^{s_1} \cdots n_k^{s_k}} = \sum_{n_1 > n_2 > \cdots > n_k > 0} \ \prod_{i=1}^k \frac{1}{n_i^{s_i}}, \!\]

$w=s_1+\cdots+s_k$를 weight, $k$를 depth로 부른다

이중 제타

오일러의 공식

$$\zeta(2,1)=\zeta(3)$$

테이블

$$ \begin{array}{c|c} \zeta (2,1) & 1.202056903 \\ \zeta (2,2) & 0.8117424253 \\ \zeta (2,3) & 0.7115661976 \\ \zeta (2,4) & 0.6745239140 \\ \zeta (2,5) & 0.6587533876 \\ \zeta (2,6) & 0.6515651637 \\ \zeta (3,1) & 0.2705808084 \\ \zeta (3,2) & 0.2288103976 \\ \zeta (3,3) & 0.2137988682 \\ \zeta (3,4) & 0.2075050146 \\ \zeta (3,5) & 0.2046611370 \\ \zeta (3,6) & 0.2033231925 \\ \zeta (4,1) & 0.09655115999 \\ \zeta (4,2) & 0.08848338245 \\ \zeta (4,3) & 0.08515982253 \\ \zeta (4,4) & 0.08367311302 \\ \zeta (4,5) & 0.08297782969 \\ \zeta (4,6) & 0.08264426276 \\ \zeta (5,1) & 0.04053689727 \\ \zeta (5,2) & 0.03857512434 \\ \zeta (5,3) & 0.03770767298 \\ \zeta (5,4) & 0.03730477856 \\ \zeta (5,5) & 0.03711229713 \\ \end{array} $$

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,

$$\zeta(r)\zeta(s)=\sum_{a=0}^{s-1}\binom{a+r-1}{a}\zeta(r+a,s-a)+\sum_{a=0}^{r-1}\binom{a+s-1}{a}\zeta(s+a,r-a)$$

예

$$ \zeta(2) \zeta(3)=\zeta(2,3)+3\zeta(3,2)+6 \zeta(4,1) $$

기타

다음이 성립한다

$$ 2\zeta(n,1)=n \zeta(n+1)-\sum_{i=1}^{n-2}\zeta(n-i)\zeta(i+1) $$

예

$$ \begin{align} \zeta(2,1)&=\zeta(3) \\ 2\zeta(3,1)&=-\zeta(2)^2+3\zeta(4) \\ \zeta(4,1)&=2\zeta(5)-\zeta(2)\zeta(3) \\ 2\zeta(5,1)&=-\zeta(3)^2-2\zeta(2)\zeta(4)+5\zeta(6) \end{align} $$

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.

테이블

추측

주어진 무게 $s$의 다중 제타 값으로 생성되는 $\mathbb{Q}$-벡터 공간은 다음과 같은 기저를 갖는다 \begin{array}{c|c} s & \\ \hline 2 & \zeta_2 \\ 3 & \zeta_3 \\ 4 & \zeta_2^2 \\ 5 & \zeta_5,\zeta_2 \zeta_3 \\ 6 & \zeta_3^2,\zeta_2^3 \\ 7 & \zeta_7,\zeta_2 \zeta_5,\zeta_2^2 \zeta_3 \\ 8 & \zeta_{5,3},\zeta_3 \zeta_5,\zeta_2 \zeta_3^2,\zeta_2^4 \\ 9 & \zeta_9,\zeta_2 \zeta_7,\zeta_2^2 \zeta_5,\zeta_3^3,\zeta_2^3 \zeta_3 \\ 10 & \zeta_{7,3},\zeta_2 \zeta_{5,3},\zeta_3 \zeta_7,\zeta_5^2,\zeta_2 \zeta_3 \zeta_5,\zeta_2^2 \zeta_3^2,\zeta_2^5 \\ \end{array}

메모

계산 리소스

https://docs.google.com/file/d/0B8XXo8Tve1cxU0NmSDg0cGNyYmM/edit
https://oeis.org/A000931
Multisum
Anzai, C., and Y. Sumino. ‘Algorithms to Evaluate Multiple Sums for Loop Computations’. arXiv:1211.5204 [hep-Ph, Physics:hep-Th, Physics:math-Ph], 22 November 2012. http://arxiv.org/abs/1211.5204.
- http://www.tuhep.phys.tohoku.ac.jp/~program/

사전 형태의 자료

http://en.wikipedia.org/wiki/Multiple_zeta_function

리뷰, 에세이, 강의노트

Schneps, Leila. ‘ARI, GARI, Zig and Zag: An Introduction to Ecalle’s Theory of Multiple Zeta Values’. arXiv:1507.01534 [math], 6 July 2015. http://arxiv.org/abs/1507.01534.
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