Holmstrom Multiple zeta values

arXiv:1101.1594 Multiple Dedekind Zeta Functions from arXiv Front: math.NT by Ivan Horozov In this paper we define multiple Dedekind zeta functions as a new type of iterated integrals. This new class of functions have many properties. They can be written as infinite sums. They have integral representation and meromorphic continuation. We also prove that at the positive integers the multiple Dedekind zeta values are periods. In particular, the Dedekind zeta function can be written in terms of the new type of iterated integral and at the positive integers the corresponding iterated integrals give periods.

arXiv:0911.2643 Multizeta values: Lie algebras and periods on $\mathfrak{M}_{0,n}$ from arXiv Front: math.NT by Sarah Carr (there are also other things by Carr) This thesis is a study of algebraic and geometric relations between multizeta values. In chapter 2, we prove a result which gives the dimension of the associated depth-graded pieces of the double shuffle Lie algebra in depths 1 and 2. In chapters 3 and 4, we study geometric relations between multizeta values coming from their expression as periods on $\mathfrak{M}_{0,n}$. The key ingredient in this study is the top dimensional de Rham cohomology of special partially compactified moduli spaces associated to multizeta values. In chapter 3, we give an explicit expression for a basis, represented by polygons, of this cohomology. In chapter 4, we generalize this method to explicitly describe the bases of the cohomology of other partially compactified moduli spaces. This thesis concludes with a result which gives a new presentation of $Pic(\overline{\mathfrak{M}}_{0,n})$.

arXiv:0907.2557 The Multiple Zeta Value Data Mine from arXiv Front: math.NT by J. Blümlein, D. J. Broadhurst, J. A. M. Vermaseren We provide a data mine of proven results for multiple zeta values (MZVs) of the form $\zeta(s_1,s_2,…,s_k)=\sum_{n_1>n_2>…>n_k>0}^\infty {1/(n_1^{s_1}

… n_k^{s_k})}$with weight$w=\sum_{i=1}^k s_i$and depth$k$and for Euler sums of the form$\sum_{n_1>n_2>…>n_k>0}^\infty t{(\epsilon_1^{n_1} …\epsilon_1 ^{n_k})/ (n_1^{s_1} … n_k^{s_k}) }$with signs$\epsilon_i=\pm1$. Notably, we achieve explicit proven reductions of all MZVs with weights$w\le22$, and all Euler sums with weights$w\le12$, to bases whose dimensions, bigraded by weight and depth, have sizes in precise agreement with the Broadhurst--Kreimer and Broadhurst conjectures. Moreover, we lend further support to these conjectures by studying even greater weights ($w\le30$), using modular arithmetic. To obtain these results we derive a new type of relation for Euler sums, the Generalized Doubling Relations. We elucidate the “pushdown” mechanism, whereby the ornate enumeration of primitive MZVs, by weight and depth, is reconciled with the far simpler enumeration of primitive Euler sums. There is some evidence that this pushdown mechanism finds its origin in doubling relations. We hope that our data mine, obtained by exploiting the unique power of the computer algebra language {\sc form}, will enable the study of many more such consequences of the double-shuffle algebra of MZVs, and their Euler cousins, which are already the subject of keen interest, to practitioners of quantum field theory, and to mathematicians alike.

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