다중 제타 값

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,

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

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

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

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} \]

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, January 2012, http://www.math.ias.edu/files/deligne/012312MultiZetas.pdf

노트

말뭉치

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.^[1]
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.^[2]
For more details see the paper "Relations of multiple zeta values and their algebraic expression".^[2]
See the talk "Algebraic structures on the set of multiple zeta values" for one approach to this problem.^[2]
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).^[2]
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.^[3]
Classical theory of multiple zeta values 1.1.^[4]
Denition of multiple zeta values 1.3.^[4]
Integral representation of multiple zeta values 1.6.^[4]
Two families of motivic multiple zeta values and Zagiers theorem 5.4.^[4]
Where zeta functions appear in physics as expressions for vacuum amplitudes, so multiple zeta functions appear in expressions for more general scattering amplitudes.^[5]
The intricate combinatorics of these becomes often more tractable when re-expressing them as motivic multiple zeta values.^[5]
It turns out that multiple zeta values are also related to modular forms but that here the relationship is much more mysterious.^[6]
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.^[6]
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.^[6]
In particular, the study of elliptic multiple zeta values should offer a more conceptual explanation of the Broadhurst-Kreimer conjecture.^[6]
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.^[7]
Finally, we apply the algebraic approach to relations of finite partial sums of multiple zeta value series.^[7]
The multiple zeta value (a1, a2, . . . , ak), with ai positive integers, is dened by (a1, a2, . .^[8]
The sum (cid:80)k i=1 ai of the ai is called weight of the multiple zeta value.^[8]
I now want to convince you that multiple zeta values satisfy a huge number of relations.^[8]
Weve already seen how the product of two RZVs can be written as a combination of multiple zeta values by multiplying the dening sums.^[8]
For , let be the sum of all multiple zeta values with even arguments whose weight is and whose depth is .^[9]
The multiple zeta functions have attracted considerable interest in recent years.^[9]
Derivation and extended double shue (EDS) relations for multiple zeta values (MZVs) are proved.^[10]
In recent years, there has been a considerable amount of interest in certain real numbers called multiple zeta values (MZVs).^[10]
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.^[11]
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.^[11]
Normally multiple zeta values (MZVs for short) mean the former and are denoted by (k).^[12]
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.^[12]
Multiple zeta values and functions are defined by the following nested infinite series.^[13]
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.^[13]
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.^[13]
We study a variant of multiple zeta values of level 2, which forms a subspace of the space of alternating multiple zeta values.^[14]
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.^[14]
We also give some conjectures on relations between our values, Homans values, and multiple zeta values.^[14]
In this paper, we study the following variant of the multiple zeta value, 1.^[14]
Values of Euler-Zagiers multiple zeta function at non-positive integers are studied, especially at (0, 0, . . . , n) and (n, 0, . .^[15]
Zagiers multiple zeta function dened by (cid:88) (2) k(s1, s2, . .^[15]
We will continue this study of multiple zeta values at non-positive inte- gers in detail.^[15]
We dene regular values of multiple zeta function at non-positive integers by (3) k(r1, r2, . . .^[15]
Multiple zeta values came out as revitalization of the zeta and double zeta values 200 years later.^[16]
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.^[16]
We call the above number a multiple zeta value of depth and weight .^[16]
Abstract The sum formula is one of the most well-known relations among multiple zeta values.^[17]
This paper proves a conjecture of Kaneko predicting that an analogous formula holds for finite multiple zeta values.^[17]
Multiple zeta values have been studied by a wide variety of methods.^[18]
Many relations are known between multiple zeta values (k1, . . .^[19]
A relation coming from the associator condition for the Drinfeld associator, the generating function of multiple zeta values, is a geometric relation.^[19]
By the theory of mixed motives, we can control the dimension of the rational linear hull of multiple zeta values.^[19]
We dene a multiple zeta value (k1, . . .^[19]
This is the function eld analog of the Euler-Zagier multiple zeta function d(s1, . . .^[20]
He then dened the multiple zeta values of depth d by d(a1, . . .^[20]
In this paper we initiate the study of multiple zeta values over global function elds.^[20]
Nonetheless, we believe that the multiple zeta function Zd (K; s1, . .^[20]
On generating functions of multiple zeta values and generalized hypergeometric functions, Manuscr.^[21]
Some particular multiple zeta values and sums of multiple zeta values can be further expressed as double integrals.^[22]
Key words and phrases: multiple zeta value; duality theorem; sum formula.^[22]
Note that the special case q = 1 of (1.2) then gives the sum formula of multiple zeta values.^[22]
In this paper, we add further factors to multiple zeta values with parameters and produce the following results.^[22]

소스

↑ Multiple zeta function
↑ ^2.0 ^2.1 ^2.2 ^2.3 Multiple Zeta Values
↑ Updated mars 2, 2017
↑ ^4.0 ^4.1 ^4.2 ^4.3 Clay mathematics proceedings
↑ ^5.0 ^5.1 multiple zeta values in nLab
↑ ^6.0 ^6.1 ^6.2 ^6.3 Multiple zeta values and modular forms
↑ ^7.0 ^7.1 Algebraic Aspects of Multiple Zeta Values
↑ ^8.0 ^8.1 ^8.2 ^8.3 Multiple zeta values
↑ ^9.0 ^9.1 Algorithms for Some Euler-Type Identities for Multiple Zeta Values
↑ ^10.0 ^10.1 Compositio math. 142 (2006) 307–338
↑ ^11.0 ^11.1 Multiple zeta values, Padé approximation and Vasilyev's conjecture
↑ ^12.0 ^12.1 (cid:1)
↑ ^13.0 ^13.1 ^13.2 Multiple zeta values and functions
↑ ^14.0 ^14.1 ^14.2 ^14.3 On multiple zeta values of level two
↑ ^15.0 ^15.1 ^15.2 ^15.3 Multiple zeta values at non-positive integers
↑ ^16.0 ^16.1 ^16.2 An introduction to the Riemann zeta function, multiple zeta function, multiple zeta values (MZV or Euler-Zagier sums) and some of their special evaluations
↑ ^17.0 ^17.1 Sum formula for finite multiple zeta values
↑ [PDF Algebraic Aspects of Multiple Zeta Values]
↑ ^19.0 ^19.1 ^19.2 ^19.3 Geometry of multiple zeta values
↑ ^20.0 ^20.1 ^20.2 ^20.3 Proceedings of symposia in pure mathematics
↑ Multiple zeta functions and polylogarithms over global function fields
↑ ^22.0 ^22.1 ^22.2 ^22.3 New zealand journal of mathematics

메타데이터

위키데이터

ID : Q2523239

Spacy 패턴 목록

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

[ref_a5b14997-1] Multiple zeta function

[ref_3343c3dc-2] 2.0 ^2.1 ^2.2 ^2.3 Multiple Zeta Values

[ref_bb0d4941-3] Updated mars 2, 2017

[ref_af2f7b9f-4] 4.0 ^4.1 ^4.2 ^4.3 Clay mathematics proceedings

[ref_65de1e12-5] 5.0 ^5.1 multiple zeta values in nLab

[ref_f109bda9-6] 6.0 ^6.1 ^6.2 ^6.3 Multiple zeta values and modular forms

[ref_6182652b-7] 7.0 ^7.1 Algebraic Aspects of Multiple Zeta Values

[ref_1d34ad17-8] 8.0 ^8.1 ^8.2 ^8.3 Multiple zeta values

[ref_ac4a1a73-9] 9.0 ^9.1 Algorithms for Some Euler-Type Identities for Multiple Zeta Values

[ref_28713f3b-10] 10.0 ^10.1 Compositio math. 142 (2006) 307–338

[ref_c00a97bd-11] 11.0 ^11.1 Multiple zeta values, Padé approximation and Vasilyev's conjecture

[ref_21ade28d-12] 12.0 ^12.1 (cid:1)

[ref_9a86a6a3-13] 13.0 ^13.1 ^13.2 Multiple zeta values and functions

[ref_0cb0d01a-14] 14.0 ^14.1 ^14.2 ^14.3 On multiple zeta values of level two

[ref_b2d96d20-15] 15.0 ^15.1 ^15.2 ^15.3 Multiple zeta values at non-positive integers

[ref_e27aa0a9-16] 16.0 ^16.1 ^16.2 An introduction to the Riemann zeta function, multiple zeta function, multiple zeta values (MZV or Euler-Zagier sums) and some of their special evaluations

[ref_62cd260d-17] 17.0 ^17.1 Sum formula for finite multiple zeta values

[ref_a1785ff5-18] [PDF Algebraic Aspects of Multiple Zeta Values]

[ref_e4877657-19] 19.0 ^19.1 ^19.2 ^19.3 Geometry of multiple zeta values

[ref_0bdf5974-20] 20.0 ^20.1 ^20.2 ^20.3 Proceedings of symposia in pure mathematics

[ref_38a2257a-21] Multiple zeta functions and polylogarithms over global function fields

[ref_6d667b08-22] 22.0 ^22.1 ^22.2 ^22.3 New zealand journal of mathematics

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