Evaluations of $k$-fold Euler/Zagier sums: a compendium of results for arbitrary $k$

Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle product rule allows the possibility of a combinatorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with certain repeated arguments. We also prove a similar cyclic sum identity. Finally, we present extensive computational evidence supporting an infinite family of conjectured MZV identities that simultaneously generalize the Zagier identity.

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“…which was proven in [5,1], gives 4 n ζ({3, 1} n ) = π 4n n r=−n (−1) r (2n − 2r + 1)! (2n + 2r + 1)!…”

Section: Proof Of the Zagier Conjecturementioning

confidence: 80%

“…For n = 0, (17) trivially reduces to the known evaluation (14). If all m i 's are equal, (17) specializes to conjecture (18) of [1].…”

Section: Conjectured Generalizations Of the Zagier Identitymentioning

confidence: 99%

Combinatorial Aspects of Multiple Zeta Values

Borwein¹,

Bradley²,

Broadhurst³

et al. 1998

Electron. J. Combin.

100

131

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“…In section 3, we mentioned recent research on multiple zeta values, which play a key role in quantum field theory [12]. More generally, one may define Euler sums by [8] ζ s 1 , s 2 · · · s r σ 1 , σ 2 · · · σ r :=…”

Section: Reduction Of Euler Sumsmentioning

confidence: 99%

Parallel integer relation detection: Techniques and applications

2000

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Let {x 1 , x 2 , · · · , x n } be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers a k , if they exist, such that a 1 x 1 + a 2 x 2 + · · · + a n x n = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing. This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Single-and multi-level implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in number theory, quantum field theory and chaos theory. IntroductionLet x = (x 1 , x 2 , · · · , x n ) be a vector of real numbers. x is said to possess an integer relation if there exist integers a i , not all zero, such that a 1 x 1 + a 2 x 2 + · · · + a n x n = 0. By an integer relation algorithm, we mean a practical computational scheme that can recover (provided the computer implementation has sufficient numeric precision) the vector of integers a i , if it exists, or can produce bounds within which no integer relation exists.The problem of finding integer relations among a set of real numbers was first studied by Euclid, who gave an iterative scheme which, when applied to two real numbers, either terminates, yielding an exact relation, or produces an infinite sequence of approximate relations. The generalization of this problem for n > 2 was attempted by Euler, Jacobi, Poincaré, Minkowski, Perron, Brun, Bernstein, among others. The first integer relation algorithm with the required properties mentioned above was discovered in 1977 by Ferguson and Forcade [18]. Since then, a number of other integer relation algorithms have been discovered, including the "HJLS" algorithm [19] (which is based on the LLL algorithm), and the "PSLQ" algorithm.

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“…However, we have not been able to locate in the literature an exact equivalent of our length m identities (4.25),(4.28) 3 . The only identities available for arbitrary lengths and levels are those based on the 'shuffle algebra' [6,26,10,7,8] and its generalizations [23]. Of those the 'depth-length' shuffle identities (which are also called 'stuffle identities' or ' * products' [22]) are obviously related, and in fact equivalent to our 'permutation' identities, as we have convinced ourselves.…”

Section: Discussionmentioning

confidence: 88%

“…Further results for the length two case can be found in [42,1]. Systematic investigations of Euler-Zagier sums of length higher than two have been undertaken only in recent years [34,21,30,24,20,9,22,6,31,2,25]. From the point of view of physics the study of their relations is relevant for attempts at a classification of the possible ultraviolet divergences in quantum field theory [12,13].…”

Section: Introductionmentioning

confidence: 99%

A quantum field theoretical representation of Euler‐Zagier sums

Müller¹,

Schubert

2002

International Journal of Mathematics and Mathematical Sciences

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We establish a novel representation of arbitrary Euler-Zagier sums in terms of weighted vacuum graphs. This representation uses a toy quantum field theory with infinitely many propagators and interaction vertices. The propagators involve Bernoulli polynomials and Clausen functions to arbitrary orders. The Feynman integrals of this model can be decomposed in terms of a vertex algebra whose structure we investigate. We derive a large class of relations between multiple zeta values, of arbitrary lengths and weights, using only a certain set of graphical manipulations on Feynman diagrams. Further uses and possible generalizations of the model are pointed out.

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Evaluations of $k$-fold Euler/Zagier sums: a compendium of results for arbitrary $k$

Cited by 156 publications

References 12 publications

Combinatorial Aspects of Multiple Zeta Values

Combinatorial Aspects of Multiple Zeta Values

Parallel integer relation detection: Techniques and applications

A quantum field theoretical representation of Euler‐Zagier sums

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