# 다중 제타 값

둘러보기로 가기 검색하러 가기

## 개요

• 리만제타함수의 다변수 일반화 $$\zeta(s_1, \ldots, s_k)$$
• $$s_i$$ 가 양의 정수일 때, 오일러 합이라 불림
• 정수론의 중요한 주제
• 물리에서 산란 진폭 등의 계산에서 등장

## 정의

• $$s_1>1, \cdots, s_k$$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

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

## 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,

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

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

### 기타

• 다음이 성립한다

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

\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}

## 다중 제타 값의 공간

• 주어진 무게를 갖는 다중 제타 값이 이루는 유리수체 위에서 정의된 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 한다
• 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}

## 테이블

### 이중 제타 값

$\begin{array}{c|c|c} \zeta (2,1) & 1.2021 & \zeta (3) \\ \zeta (2,2) & 0.81174 & \frac{\pi ^4}{120} \\ \zeta (2,3) & 0.71157 & \frac{9 \zeta (5)}{2}-\frac{\pi ^2 \zeta (3)}{3} \\ \zeta (2,4) & 0.67452 & \frac{5 \pi ^6}{2268}-\zeta (3)^2 \\ \zeta (2,5) & 0.65875 & -\frac{\pi ^4 \zeta (3)}{45}+10 \zeta (7)-\frac{2 \pi ^2 \zeta (5)}{3} \\ \zeta (2,6) & 0.65157 & \frac{\pi ^8}{14175}-\zeta (6,2) \\ \zeta (2,7) & 0.64817 & -\frac{1}{945} 2 \pi ^6 \zeta (3)-\pi ^2 \zeta (7)+\frac{35 \zeta (9)}{2}-\frac{2 \pi ^4 \zeta (5)}{45} \\ \zeta (3,1) & 0.27058 & \frac{\pi ^4}{360} \\ \zeta (3,2) & 0.22881 & \frac{1}{2} \left(\pi ^2 \zeta (3)-11 \zeta (5)\right) \\ \zeta (3,3) & 0.21380 & \frac{\zeta (3)^2}{2}-\frac{\pi ^6}{1890} \\ \zeta (3,4) & 0.20751 & \frac{\pi ^4 \zeta (3)}{90}+\frac{5 \pi ^2 \zeta (5)}{3}-18 \zeta (7) \\ \zeta (3,5) & 0.20466 & -4 \zeta (3) \zeta (5)+\frac{5 \zeta (6,2)}{2}+\frac{41 \pi ^8}{75600} \\ \zeta (3,6) & 0.20332 & \frac{\pi ^6 \zeta (3)}{945}+\frac{\pi ^4 \zeta (5)}{15}+\frac{7 \pi ^2 \zeta (7)}{2}-\frac{85 \zeta (9)}{2} \\ \zeta (4,1) & 0.096551 & 2 \zeta (5)-\frac{\pi ^2 \zeta (3)}{6} \\ \zeta (4,2) & 0.088483 & \zeta (3)^2-\frac{4 \pi ^6}{2835} \\ \zeta (4,3) & 0.085160 & 17 \zeta (7)-\frac{5 \pi ^2 \zeta (5)}{3} \\ \zeta (4,4) & 0.083673 & \frac{\pi ^8}{113400} \\ \zeta (4,5) & 0.082978 & \frac{1}{90} \left(-4 \pi ^4 \zeta (5)-525 \pi ^2 \zeta (7)+5625 \zeta (9)\right) \\ \zeta (5,1) & 0.040537 & \frac{\pi ^6-630 \zeta (3)^2}{1260} \\ \zeta (5,2) & 0.038575 & \frac{\pi ^4 \zeta (3)}{45}+\frac{5 \pi ^2 \zeta (5)}{6}-11 \zeta (7) \\ \zeta (5,3) & 0.037708 & 5 \zeta (3) \zeta (5)-\frac{5 \zeta (6,2)}{2}-\frac{7 \pi ^8}{10800} \\ \zeta (5,4) & 0.037305 & \frac{1}{18} \left(\pi ^4 \zeta (5)+105 \pi ^2 \zeta (7)-1143 \zeta (9)\right) \\ \zeta (6,1) & 0.018356 & -\frac{\pi ^4 \zeta (3)}{90}+3 \zeta (7)-\frac{\pi ^2 \zeta (5)}{6} \\ \zeta (6,2) & 0.017820 & \zeta (6,2) \\ \zeta (6,3) & 0.017573 & -\frac{\pi ^4 \zeta (5)}{15}+\frac{83 \zeta (9)}{2}-\frac{7 \pi ^2 \zeta (7)}{2} \\ \end{array}$

## 관련논문

• Masanobu Kaneko, Shuji Yamamoto, A new integral-series identity of multiple zeta values and regularizations, arXiv:1605.03117 [math.NT], May 10 2016, http://arxiv.org/abs/1605.03117
• Benjamin Enriquez, Hidekazu Furusho, A stabilizer interpretation of double shuffle Lie algebras, arXiv:1605.02838 [math.QA], May 10 2016, http://arxiv.org/abs/1605.02838
• Johannes Broedel, Nils Matthes, Oliver Schlotterer, Relations between elliptic multiple zeta values and a special derivation algebra, arXiv:1507.02254 [hep-th], July 08 2015, http://arxiv.org/abs/1507.02254, 10.1088/1751-8113/49/15/155203, http://dx.doi.org/10.1088/1751-8113/49/15/155203, J.Phys. A49 (2016) 155203
• Zhonghua Li, Chen Qin, Stuffle product formulas of multiple zeta values, arXiv:1603.08332[math.NT], March 28 2016, http://arxiv.org/abs/1603.08332v1
• Zhonghua Li, Chen Qin, Shuffle product formulas of multiple zeta values, http://arxiv.org/abs/1603.05786v1
• Michael E. Hoffman, On Multiple Zeta Values of Even Arguments, http://arxiv.org/abs/1205.7051v3
• Ding Ma, Koji Tasaka, On triple zeta values of even weight and their connections with period polynomials, http://arxiv.org/abs/1603.01013v1
• Vieru, Andrei. “Analytic Renormalization of Multiple Zeta Functions. Geometry and Combinatorics of Generalized Euler Reflection Formula for MZV.” arXiv:1601.04703 [math], January 18, 2016. http://arxiv.org/abs/1601.04703.
• Jarossay, David. “Depth Reductions for Associators.” arXiv:1601.01161 [math], January 6, 2016. http://arxiv.org/abs/1601.01161.
• Murahara, Hideki. “Derivation Relations for Finite Multiple Zeta Values.” arXiv:1512.08696 [math], December 29, 2015. http://arxiv.org/abs/1512.08696.
• Ebrahimi-Fard, Kurusch, Dominique Manchon, and Johannes Singer. “Duality and (q-)multiple Zeta Values.” arXiv:1512.00753 [math], December 2, 2015. http://arxiv.org/abs/1512.00753.
• Zerbini, Federico. “Single-Valued Multiple Zeta Values in Genus 1 Superstring Amplitudes.” arXiv:1512.05689 [hep-Th], December 17, 2015. http://arxiv.org/abs/1512.05689.
• Panzer, Erik. “The Parity Theorem for Multiple Polylogarithms.” arXiv:1512.04482 [math], December 14, 2015. http://arxiv.org/abs/1512.04482.
• Singer, Johannes, and Jianqiang Zhao. “Finite and Symmetrized Colored Multiple Zeta Values.” arXiv:1512.03691 [math], December 11, 2015. http://arxiv.org/abs/1512.03691.
• Ebrahimi-Fard, Kurusch, Dominique Manchon, Johannes Singer, and Jianqiang Zhao. “Renormalisation Group for Multiple Zeta Values.” arXiv:1511.06720 [math], November 20, 2015. http://arxiv.org/abs/1511.06720.
• Ma, Ding. “Connections between Double Zeta Values Relative to $$\mu_N$$, Hecke Operators $$T_N$$, and Newforms of Level $$\Gamma_0(N)$$ for $$N=2,3$$.” arXiv:1511.06102 [math], November 19, 2015. http://arxiv.org/abs/1511.06102.
• Ebrahimi-Fard, Kurusch, Dominique Manchon, Johannes Singer, and Jianqiang Zhao. “Transfer Group for Renormalized Multiple Zeta Values.” arXiv:1510.09159 [math], October 30, 2015. http://arxiv.org/abs/1510.09159.
• Chang, Chieh-Yu. “Linear Relations among Double Zeta Values in Positive Characteristic.” arXiv:1510.06519 [math], October 22, 2015. http://arxiv.org/abs/1510.06519.
• Dotzel, Michael, and Ivan Horozov. “Shuffle Product for Multiple Dedekind Zeta Values over Imaginary Quadratic Fields.” arXiv:1509.08400 [math], September 28, 2015. http://arxiv.org/abs/1509.08400.
• Matthes, Nils. “Elliptic Double Zeta Values.” arXiv:1509.08760 [math], September 29, 2015. http://arxiv.org/abs/1509.08760.
• Furusho, Hidekazu, Yasushi Komori, Kohji Matsumoto, and Hirofumi Tsumura. “Desingularization of Complex Multiple Zeta-Functions.” arXiv:1508.06920 [math], August 27, 2015. http://arxiv.org/abs/1508.06920.
• Broedel, Johannes, Nils Matthes, and Oliver Schlotterer. ‘Relations between Elliptic Multiple Zeta Values and a Special Derivation Algebra’. arXiv:1507.02254 [hep-Th], 8 July 2015. http://arxiv.org/abs/1507.02254.
• Oyama, Kojiro. ‘Ohno’s Relation for Finite Multiple Zeta Values’. arXiv:1506.00833 [math], 2 June 2015. http://arxiv.org/abs/1506.00833.
• Kaneko, Masanobu, and Mika Sakata. ‘On Multiple Zeta Values of Extremal Height’. arXiv:1505.01014 [math], 5 May 2015. http://arxiv.org/abs/1505.01014.
• 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.
• Zagier, Don. ‘Evaluation of the Multiple Zeta Values zeta(2,...,2,3,2,...,2)’. Annals of Mathematics 175, no. 2 (1 March 2012): 977–1000. doi:10.4007/annals.2012.175.2.11.
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• Ihara, Kentaro, Masanobu Kaneko, and Don Zagier. ‘Derivation and Double Shuffle Relations for Multiple Zeta Values’. Compositio Mathematica 142, no. 02 (March 2006): 307–38. doi:10.1112/S0010437X0500182X.

## 노트

### 말뭉치

1. When s 1 , ..., s k are all positive integers (with s 1 > 1) these sums are often called multiple zeta values (MZVs) or Euler sums.
2. In fact, this appears to be just the simplest of a whole family of similar identities: see the paper "Combinatorial aspects of multiple zeta values" for details.
3. For more details see the paper "Relations of multiple zeta values and their algebraic expression".
4. See the talk "Algebraic structures on the set of multiple zeta values" for one approach to this problem.
5. You can play around with multiple zeta values yourself using the EZFace calculator at CECM (Centre for Experimental and Constructive Mathematics at Simon Fraser University).
6. They show that the Qvector space spanned by the multiple zeta values is in fact an algebra: a product of linear combinations of numbers of the form (s) is again a linear combination of such numbers.
7. Classical theory of multiple zeta values 1.1.
8. Denition of multiple zeta values 1.3.
9. Integral representation of multiple zeta values 1.6.
10. Two families of motivic multiple zeta values and Zagiers theorem 5.4.
11. Where zeta functions appear in physics as expressions for vacuum amplitudes, so multiple zeta functions appear in expressions for more general scattering amplitudes.
12. The intricate combinatorics of these becomes often more tractable when re-expressing them as motivic multiple zeta values.
13. It turns out that multiple zeta values are also related to modular forms but that here the relationship is much more mysterious.
14. Their observation is subsumed in the far-reaching Broadhurst-Kreimer conjecture which describes the algebraic structure of multiple zeta values entirely in terms of modular forms.
15. In light of the seemingly mysterious relationship between multiple zeta values and period integrals of modular forms, it is natural to ask for a common framework accommodating both objects.
16. In particular, the study of elliptic multiple zeta values should offer a more conceptual explanation of the Broadhurst-Kreimer conjecture.
17. We also discuss some results about multiple zeta values that can be stated in terms of derivations and cyclic derivations of Q〈x,y〉, and we define an action of QSym on Q〈x,y〉 that appears useful.
18. Finally, we apply the algebraic approach to relations of finite partial sums of multiple zeta value series.
19. The multiple zeta value (a1, a2, . . . , ak), with ai positive integers, is dened by (a1, a2, . .
20. The sum (cid:80)k i=1 ai of the ai is called weight of the multiple zeta value.
21. I now want to convince you that multiple zeta values satisfy a huge number of relations.
22. Weve already seen how the product of two RZVs can be written as a combination of multiple zeta values by multiplying the dening sums.
23. For , let be the sum of all multiple zeta values with even arguments whose weight is and whose depth is .
24. The multiple zeta functions have attracted considerable interest in recent years.
25. Derivation and extended double shue (EDS) relations for multiple zeta values (MZVs) are proved.
26. In recent years, there has been a considerable amount of interest in certain real numbers called multiple zeta values (MZVs).
27. It is based on a simultaneous Padé approximation problem involving certain multiple polylogarithms, which evaluated at the point 1 are multiple zeta values equal to powers of pi.
28. At the point 1, this provides a rational linear combination of 1 and multiple zeta values in an extended sense that turn out to be values of the Riemann zeta function at odd integers.
29. Normally multiple zeta values (MZVs for short) mean the former and are denoted by (k).
30. They proved that the special values of k(s) at non-positive integers are given by poly-Bernoulli numbers and the values at positive integers are given in terms of multiple zeta values.
31. Multiple zeta values and functions are defined by the following nested infinite series.
32. When the arguments are all positive integers, this is called the multiple zeta value; on the other hand, as a function of complex variables, this is called the multiple zeta function.
33. Further, we develop an analytic and p-adic theory of multiple zeta functions, and with the theory of multiple zeta values, we attempt to contribute a new development in the area of multiple zetas.
34. We study a variant of multiple zeta values of level 2, which forms a subspace of the space of alternating multiple zeta values.
35. This variant, which is regarded as the shue counterpart of Homans odd variant, exhibits nice properties such as duality, shue product, parity results, etc., like ordinary multiple zeta values.
36. We also give some conjectures on relations between our values, Homans values, and multiple zeta values.
37. In this paper, we study the following variant of the multiple zeta value, 1.
38. Values of Euler-Zagiers multiple zeta function at non-positive integers are studied, especially at (0, 0, . . . , n) and (n, 0, . .
39. Zagiers multiple zeta function dened by (cid:88) (2) k(s1, s2, . .
40. We will continue this study of multiple zeta values at non-positive inte- gers in detail.
41. We dene regular values of multiple zeta function at non-positive integers by (3) k(r1, r2, . . .
42. Multiple zeta values came out as revitalization of the zeta and double zeta values 200 years later.
43. In 1992, Michael Hoffman and Don Zagier generalized the concept of Riemann zeta function & values of the zeta function to the multiple zeta function and multiple zeta values.
44. We call the above number a multiple zeta value of depth and weight .
45. Abstract The sum formula is one of the most well-known relations among multiple zeta values.
46. This paper proves a conjecture of Kaneko predicting that an analogous formula holds for finite multiple zeta values.
47. Multiple zeta values have been studied by a wide variety of methods.
48. Many relations are known between multiple zeta values (k1, . . .
49. A relation coming from the associator condition for the Drinfeld associator, the generating function of multiple zeta values, is a geometric relation.
50. By the theory of mixed motives, we can control the dimension of the rational linear hull of multiple zeta values.
51. We dene a multiple zeta value (k1, . . .
52. This is the function eld analog of the Euler-Zagier multiple zeta function d(s1, . . .
53. He then dened the multiple zeta values of depth d by d(a1, . . .
54. In this paper we initiate the study of multiple zeta values over global function elds.
55. Nonetheless, we believe that the multiple zeta function Zd (K; s1, . .
56. On generating functions of multiple zeta values and generalized hypergeometric functions, Manuscr.
57. Some particular multiple zeta values and sums of multiple zeta values can be further expressed as double integrals.
58. Key words and phrases: multiple zeta value; duality theorem; sum formula.
59. Note that the special case q = 1 of (1.2) then gives the sum formula of multiple zeta values.
60. In this paper, we add further factors to multiple zeta values with parameters and produce the following results.

## 메타데이터

### Spacy 패턴 목록

• [{'LOWER': 'multiple'}, {'LOWER': 'zeta'}, {'LEMMA': 'value'}]
• [{'LOWER': 'multiple'}, {'LOWER': 'zeta'}, {'LEMMA': 'function'}]