Article

# Derivation and double shuffle relations for multiple zeta values

Compositio Mathematica (Impact Factor: 1.02). 03/2006; 142(02). DOI: 10.1112/S0010437X0500182X

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**ABSTRACT:**In their seminal paper ``Double zeta values and modular forms'' Gangl, Kaneko and Zagier defined a double Eisenstein series and used it to study the relations between double zeta values. One of their key ideas is to study the formal double space and apply the double shuffle relations. They also proved the double shuffle relations for the double Eisenstein series. More recently, Kaneko and Tasaka extended the double Eisenstein series to level 2, proved its double shuffle relations and studied the double zeta values of level 2. Motivated by the above works, we define in this paper the corresponding objects at higher levels and prove that the double Eisenstein series of level N satisfies the double shuffle relations for every positive integer N. In order to obtain our main theorem we prove a key result on the multiple divisor functions of level N and then use it to solve a complicated under-determined system of linear equations by some standard techniques from linear algebra.01/2014; - [Show abstract] [Hide abstract]

**ABSTRACT:**The Broadhurst-Kreimer (BK) conjecture describes the Hilbert series of a bigraded Lie algebra A related to the multizeta values. Brown proposed a conjectural description of the homology of this Lie algebra (homological conjecture (HC)), and showed it implies the BK conjecture. We show that a part of HC is equivalent to a presentation of A, and that the remaining part of HC is equivalent to a weaker statement. Finally, we prove that granted the first part of HC, the remaining part of HC is equivalent to either of the following equivalent statements: (a) the vanishing of the third homology group of a Lie algebra with quadratic presentation, constructed out of the period polynomials of modular forms; (b) the koszulity of the enveloping algebra of this Lie algebra.07/2014; - [Show abstract] [Hide abstract]

**ABSTRACT:**We exhibit the double q-shuffle structure for the qMZVs recently introduced by Y. Ohno, J. Okuda and W. Zudilin.10/2013;

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