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55
Special Values of Multiple Polylogarithms
 Sém. Bourbaki, 53 e année, 2000–2001, n ◦ 885, Mars 2001; Astéisque 282 (2002
"... Abstract. Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and highenergy physics. More recen ..."
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Cited by 59 (18 self)
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Abstract. Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and highenergy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier. 1.
Parallel Integer Relation Detection: Techniques and Applications
 Mathematics of Computation
, 2000
"... Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and phy ..."
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Cited by 47 (35 self)
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Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 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. Singleand multilevel 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 mathematical number theory, quantum field theory and chaos theory.
Combinatorics of (perturbative) quantum field theory
, 2000
"... We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic oper ..."
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Cited by 30 (8 self)
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We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic operation of graph insertions. Particular emphasis is given to the connection with the Riemann–Hilbert problem. Finally, we outline how these structures relate to the numbers which we see in Feynman diagrams.
Combinatorial aspects of multiple zeta values
 Electr. J. Comb
, 1998
"... 1 the electronic journal of combinatorics 5 (1998), #R38 2 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 p ..."
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Cited by 29 (7 self)
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1 the electronic journal of combinatorics 5 (1998), #R38 2 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. 1
Some examples of Mahler measures as multiple polylogarithms
 J. Number Theory
"... The Mahler measures of certain polynomials of up to five variables are given in terms of multiple polylogarithms. Each formula is homogeneous and its weight coincides with the number of variables of the corresponding polynomial. Key words: Mahler measure, Lfunctions, polylogarithms, hyperlogarithms ..."
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Cited by 23 (15 self)
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The Mahler measures of certain polynomials of up to five variables are given in terms of multiple polylogarithms. Each formula is homogeneous and its weight coincides with the number of variables of the corresponding polynomial. Key words: Mahler measure, Lfunctions, polylogarithms, hyperlogarithms, polynomials, Jensen’s formula
Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)
, 1998
"... ..."
Central Binomial Sums and Multiple Clausen Values (with Connections to Zeta Values
"... We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of nonalternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of al ..."
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Cited by 22 (9 self)
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We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of nonalternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. In the nonalternating case, there is a strong connection to polylogarithms of the sixth root of unity, encountered in the 3loop Feynman diagrams of hepth/9803091 and subsequently in hepph/9910223, hepph/9910224, condmat/9911452 and hepth/0004010.
Multiple qzeta values
 J. Algebra
"... Abstract. We introduce a qanalog of the multiple harmonic series commonly referred to as multiple zeta values. The multiple qzeta values satisfy a qstuffle multiplication rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values. Add ..."
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Cited by 22 (2 self)
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Abstract. We introduce a qanalog of the multiple harmonic series commonly referred to as multiple zeta values. The multiple qzeta values satisfy a qstuffle multiplication rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values. Additionally, multiple qzeta values can be viewed as special values of the multiple qpolylogarithm, which admits a multiple Jackson qintegral representation whose limiting case is the Drinfel’d simplex integral for the ordinary multiple polylogarithm when q = 1. The multiple Jackson qintegral representation for multiple qzeta values leads to a second multiplication rule satisfied by them, referred to as a qshuffle. Despite this, it appears that many numerical relations satisfied by ordinary multiple zeta values have no interesting qextension. For example, a suitable qanalog of Broadhurst’s formula for ζ({3, 1} n), if one exists, is likely to be rather complicated. Nevertheless, we show that a number of infinite classes of relations, including Hoffman’s partition identities, Ohno’s cyclic sum identities, Granville’s sum formula, Euler’s convolution formula, Ohno’s generalized duality relation, and the derivation relations of Ihara
Special Values of Multidimensional Polylogarithms
 TRANS. AMER. MATH. SOC
, 1998
"... Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and highenergy physics. More recently, w ..."
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Cited by 20 (12 self)
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Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and highenergy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including a longstanding conjec...
Multiple Polylogarithms: A Brief Survey
"... . We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral representation for the general multiple polylogarithm, and devel ..."
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Cited by 19 (6 self)
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. We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral representation for the general multiple polylogarithm, and develop a qanalogue of the shuffle product. 1.