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Renormalization of multiple zeta values
 J. Algebra
, 2006
"... Abstract. Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special ..."
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Abstract. Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special values of multiple zeta functions at nonpositive integers since the values are usually singular. We define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory, and following the Hopf algebra approach of Connes and Kreimer. This definition of renormalized MZVs agrees with the convergent MZVs and extends the work of IharaKanekoZagier on renormalization of MZVs with positive arguments. We further show that the important
Multiple Harmonic Sums II: Finiteness of pDivisible Sets ∗
, 2005
"... Abstract. In this paper we continue to study the multiple harmonic sums which are partial sums of multiple zeta value series (abbreviated as MZV series). We conjecture that for any prime p and any MZV series there is always some N such that if n> N then p does not divide the numerator of the nth par ..."
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Abstract. In this paper we continue to study the multiple harmonic sums which are partial sums of multiple zeta value series (abbreviated as MZV series). We conjecture that for any prime p and any MZV series there is always some N such that if n> N then p does not divide the numerator of the nth partial sum of the MZV series. This generalizes a conjecture of Eswarathasan and Levine and Boyd for harmonic series. We provide a lot of evidence for this general conjecture and make some heuristic argument to support it.
Multiple Harmonic Sums I: Generalizations of Wolstenholme’s Theorem
, 2005
"... In this note we will study the pdivisibility of multiple harmonic sums. In particular we provide some generalizations of the classical Wolstenholme’s Theorem to both homogeneous and nonhomogeneous sums. We make a few conjectures at the end of the paper and provide some very convincing evidence. ..."
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In this note we will study the pdivisibility of multiple harmonic sums. In particular we provide some generalizations of the classical Wolstenholme’s Theorem to both homogeneous and nonhomogeneous sums. We make a few conjectures at the end of the paper and provide some very convincing evidence.
CONGRUENCES OF ALTERNATING MULTIPLE HARMONIC SUMS
, 909
"... Abstract. In this sequel to [15], we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials play the key roles, in the alternating setting the Euler numbers ..."
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Abstract. In this sequel to [15], we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials play the key roles, in the alternating setting the Euler numbers and the Euler polynomials are also essential. Mathematics Subject Classification: 11M41, 11B50. 1.
Quasisymmetric functions, multiple zeta values, and rooted trees
, 2006
"... My first talk was about the algebra of multiple zeta values, and the second about Hopf algebras of rooted trees. A thread that connects the two is the Hopf algebra QSym of quasisymmetric functions. First defined by Gessel [4], QSym consists of those formal power series f ∈ Q[[t1, t2,...]] (each ti ..."
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My first talk was about the algebra of multiple zeta values, and the second about Hopf algebras of rooted trees. A thread that connects the two is the Hopf algebra QSym of quasisymmetric functions. First defined by Gessel [4], QSym consists of those formal power series f ∈ Q[[t1, t2,...]] (each ti having degree one), such that f has bounded degree, and the coefficient in f of t p1 i1 tp2 · · · tpk i2 ik equals the coefficient in f of t p1 1 tp2 2 · · · tpk k whenever i1 < i2 < · · · < ik. As a vector space, QSym is generated by the monomial quasisymmetric functions Mp1p2···pk = i1<i2<···<ik t p1 i1 tp2 · · ·tpk i2 ik. The algebra Sym of symmetric functions is a proper subalgebra of QSym: for example, M11 and M12 + M21 are symmetric, but M12 is not. As an algebra, QSym is generated by those monomial symmetric functions corresponding to Lyndon words in the positive integers [11, 6]. The subalgebra of QSym 0 ⊂ QSym generated by all Lyndon words other than M1 has the vector space basis consisting of all monomial symmetric functions Mp1p2···pk with pk> 1 (together with M ∅ = 1). There is a homorphism QSym 0 → R given by sending each ti to 1 i; that is, the monomial quasisymmetric function Mp1···pk is sent to the multiple zeta value ζ(pk, pk−1,...,p1) = i1>i2>···>ik≥1
SHUFFLE PRODUCTS FOR MULTIPLE ZETA VALUES AND PARTIAL FRACTION DECOMPOSITIONS OF ZETAFUNCTIONS OF ROOT
, 908
"... Abstract. The shuffle product plays an important role in the study of multiple zeta values. This is expressed in terms of multiple integrals, and also as a product in a certain noncommutative polynomial algebra over the rationals in two indeterminates. In this paper, we give a new interpretation of ..."
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Abstract. The shuffle product plays an important role in the study of multiple zeta values. This is expressed in terms of multiple integrals, and also as a product in a certain noncommutative polynomial algebra over the rationals in two indeterminates. In this paper, we give a new interpretation of the shuffle product. In fact, we prove that the procedure of shuffle products essentially coincides with that of partial fraction decompositions of multiple zeta values of root systems. As an application, we give a proof of extended double shuffle relations without using Drinfel’d integral expressions for multiple zeta values. Furthermore, our argument enables us to give some functional relations which include double shuffle relations.
Multiple Zeta Values and . . .
, 2006
"... We study multiple zeta values and their generalizations from the point of view of Rota–Baxter algebras. We obtain a general framework for this purpose and derive relations on multiple zeta values from relations in Rota–Baxter algebras. ..."
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We study multiple zeta values and their generalizations from the point of view of Rota–Baxter algebras. We obtain a general framework for this purpose and derive relations on multiple zeta values from relations in Rota–Baxter algebras.
Nikhef Theory Group
, 907
"... We provide a data mine of proven results for multiple zeta values (MZVs) of the form ζ(s1,s2,...,sk) = ∑ ∞ { s1 n1>n2>...>nk>0 1/(n1...nsk k)} with weight w = ∑ k i=1 si and depth k and for Euler sums of the form ∑ ∞ { n1 n1>n2>...>nk>0 (ε1...εnk 1)/(ns1 1...nsk k)} with signs εi = ±1. Notably, ..."
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We provide a data mine of proven results for multiple zeta values (MZVs) of the form ζ(s1,s2,...,sk) = ∑ ∞ { s1 n1>n2>...>nk>0 1/(n1...nsk k)} with weight w = ∑ k i=1 si and depth k and for Euler sums of the form ∑ ∞ { n1 n1>n2>...>nk>0 (ε1...εnk 1)/(ns1 1...nsk k)} with signs εi = ±1. Notably, we achieve explicit proven reductions of all MZVs with weights w ≤ 22, and all Euler sums with weights w ≤ 12, to bases whose dimensions, bigraded by weight and depth, have sizes in precise agreement with the Broadhurst–