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Jianqiang Zhao

Publications and source records attributed to Jianqiang Zhao.

82 records · Page 5Linked to original sources
arXiv2007

Double Shuffle Relations of Euler Sums

In this paper we shall develop a theory of (extended) double shuffle relations of Euler sums which generalizes that of multiple zeta values (see Ihara, Kaneko and Zagier, \emph{Derivation and double shuffle relations for multiple zeta values}. Compos. Math. \textbf{142} (2)(2006), 307--338). After setting up the general framework we provide some numerical evidence for our two main conjectures. At the end we shall prove the following long standing conjecture: for every positive integer n $$ζ(\{3\}^n)=8^nζ(\{\ol2,1\}^n).$$ The main idea is to use the double shuffle relations and the distribution relation. This particular distribution relation doesn't follow from the double shuffle relation in general. But we believe it does follow from the extended double shuffle relations.

math.NT↗
arXiv2007

An exotic shuffle relation of $ζ(\{2\}^m)$ and $ζ(\{3,1\}^n)$

In this short note we will provide a new and shorter proof of the following exotic shuffle relation of multiple zeta values: $$ζ(\{2\}^m \sha\{3,1\}^n)={2n+m\choose m} \frac{π^{4n+2m}}{(2n+1)\cdot (4n+2m+1)!}.$$ This was proved by Zagier when n=0, by Broadhurst when $m=0$, and by Borwein, Bradley, and Broadhurst when m=1. In general this was proved by Bowman and Bradley in \emph{The algebra and combinatorics of shuffles and multiple zeta values}, J. of Combinatorial Theory, Series A, Vol. \textbf{97} (1)(2002), 43--63. Our idea in the general case is to use the method of Borwein et al. to reduce the above general relation to some families of combinatorial identities which can be verified by WZ-method.

math.NT↗
arXiv2007

Renormalization of Multiple $q$-Zeta Values

In this paper we shall define the renormalization of the multiple $q$-zeta values (M$q$ZV) which are special values of multiple $q$-zeta functions $ζ_q(s_1,...,s_d)$ when the arguments are all positive integers or all non-positive integers. This generalizes the work of Guo and Zhang (math.NT/0606076v3) on the renormalization of Euler-Zagier multiple zeta values. We show that our renormalization process produces the same values if the M$q$ZVs are well-defined originally and that these renormalizations of M$q$ZV satisfy the $q$-stuffle relations if we use shifted-renormalizations for all divergent $ζ_q(s_1,...,s_d)$ (i.e., $s_1\le 1$). Moreover, when $\qup$ our renormalizations agree with those of Guo and Zhang.

math.NT↗
arXiv2003

Goncharov's relations in Bloch's higher Chow group CH^3(F,5)

Using Totaro-Bloch-Kriz's linear fractional cycles Gangl and Muller-Stach recently prove the 5-term relations for the dilogarithm in Bloch's higher Chow group CH^2(F,3) and the Kummer-Spence relations in some group G(F) over an arbitrary field F where G(F) is isomorphic to CH^3(F,5) up to torsions under the Beilinson-Soule vanishing conjecture that CH^2(F,n)=0 for n>3. In this paper we show that Goncharov's 22-term relations for the trilogarithm also hold in G(F).

math.AG↗
arXiv2003

Variations of mixed Hodge structures of multiple polylogarithms

It's well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall {\em explicitly} determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications: First we study of the limit mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple {\em poly}logarithms. Then following Deligne and Beilinson we describe an approach to defining the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. In the process we find some interesting identities relating single-valued multiple polylogarithms of the same weight $k$ when $k=2$ and 3. At the end of this paper, motivated by Zagier's conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms.

math.AG↗
arXiv2003

Analytic continuation of multiple polylogarithms

In this paper we shall define the analytic continuation of the multiple polylogarithms by using Chen's theory of iterated path integrals and compute the monodromy of all multiple logarithms explicitly.

math.AG↗